Substituting this value for C in equation (7) This equation merely states that if the instrument curve is so constructed that the abscissæ of the tracing point and the abscissæ of the curve satisfy equation (8), the tracing point of the instrument will draw an area proportional to the constant desired. To obtain this curve we will refer again to fig. 1. is the angle the arm makes with a horizontal line, L the distance between the pivot and the tracing point, H the distance of the pivot above the axis of integration of the machine, X, Y, the coordinates of the curve with reference to an axis along the axis of integration of the machine, and to one perpendicular to it through the point O, and X', Y', the coordinates of the tracing point with reference to the same axes. It will be noticed that X and x are measured from the same axis and when the machine is in operation have simultaneous values; the same may also be said of X' and x. No similar relations exist between Y and y or Y' and y', but as these terms do not appear in the essential equation, this does not matter. From the figure by assuming values of x or X, Y can be calculated and the curve fully determined. The only point not determined is what value to use for K. The smaller this value becomes, the smaller the absolute error will be, so it will pay us to use as small a value as possible. From practical considerations this value will be K = for any smaller value than this will cause 4 3 h the curve to extend outside of the tracing point. His taken as 3/2 h, for if it is taken any smaller, the results will not be very accurate, while if taken much larger, the machine tends to become unwieldy. A cosine curve may be obtained in a similar manner, going through the same set of equations and replacing the sin by the cosin. If only the odd harmonics are to be found, then it is necessary to analyze only one loop of the wave form. The integral to be evaluated is and if this is substituted in equation (1), a similar set of instrument curves may be found for this type of machine. As only one loop has to be analyzed, the instrument will be only half as large. As stated before, the essential operation consists in tracing the wave form with the intersection of the arm and instrument curve, while at the same time the derived area is drawn by the tracing point. It should be espe cially noticed that the axis of the wave form must be traced as well as the wave form itself, for otherwise no closed area would be obtained. The derived area will consist of a number of loops, the number depending on the harmonic analyzed for, and the size and shape of the wave form. Fig. 3 shows a sample wave form and derived area for the first sin harmonic. Every time the lines cross over the sign of the area changes. On all polar or rolling planimeter positive area is that area which is measured in a clockwise direction, while if the direction of rotation is reversed, the value recorded has a negative sign. This fact can be utilized in the summing up of the positive and negative areas at one reading, for if the tracing point of the planimeter follows the tracing point of the analyzer in the exact path traced by it, the tracing point of the planimeter will go around the positive loops in one direction and the negative loops in another. For this reason it is not necessary to draw out the equivalent area, but only necessary to place the tracing point of the planimeter in the tracing point of the analyzer, trace the figure, and read the planimeter directly. As the first sin or cosin loops will always be positive in area unless the ordinates themselves are negative, the tracing point of the planimeter should start at the end of this first loop, and follow the upper side around the successive loops back to the starting point. The arrows in fig. 3 illustrate this principle as applied to a derived area from a first sin harmonic analyzer. The sign of the resulting reading will tell whether the harmonic is positive or negative. The absolute error measured in inches of result for any harmonic of this machine is given by the formula .01 B where B is the maximum ordinate of the wave form to be analyzed, and h the wave length of the analyzer. Fig. 4 shows a comparison of the errors for the analyzer and for a computation method, using as a basis of comparison a constant ordinate wave form, which is nearly the only form in which errors by a computation method can be directly calculated. The lines on the figure represent conditions of equal accuracy. M in the figure represents the number of ordinates per wave length used in calculating the constants, while h represents the wave lengths of the analyzer used. This figure shows that the analyzer is more accurate, especially for the higher harmonics. Lawrence, Kansas. ART. XXXVII.-Orientite, a new hydrous Silicate of Manganese and Calcium from Cuba; by D. F. HEWETT and EARL V. SHANNON.3 CONTENTS. Introduction. Part I. Mode of Occurrence. Associated Minerals. Paragenesis. Part II. Crystallography: General character; habits; combinations; forms and angles. Physical properties: color; cleavage; hardness; specific gravity. Composition and chemical properties: analyses; pyrognostics. Introduction. Introduction.-In the course of the examination of some manganese deposits in Oriente Province, Cuba, during March and April, 1920, a crystallized silicate of manganese and calcium was discovered by D. F. Hewett. After preliminary tests, specimens were sent to E. V. Shannon, who determined that the mineral was a hydrous silicate of manganese and calcium and represented a new species. In the following statement, the crystallographic, optical and chemical studies have been made by Shannon and those with reference to mode of occurrence and genesis by Hewett. 4 As the mineral is known to occur in two localities in Oriente Province, where many manganese deposits are found, and it may be widespread in the region, it is appropriate that the geographic relation be perpetuated in the name orientite. Part I. Mode of Occurrence. The province of Oriente roughly coincides with a broad structural trough in the rocks, 1 Published by permission of the Director of the U. S. Geological Survey and the Secretary of the Smithsonian Institution. 2 4 Geologist, U. S. Geological Survey. Assistant Curator, Department of Geology, U. S. National Museum. Hayes, C. W., Vaughan, T. W., Spencer, A. C., Report on a geological reconnaissance of Cuba made under direction of Gen. Leonard Wood, 1901. Burchard, E. F., Manganese-Ore Deposits in Cuba, Trans. Am. Inst. Min. & Met. Eng., vol. 63, p. 51-104, 1920. |