GRAVITATIONAL STABILITY OF THE EARTH. 235 



We sum first with respect to m ; but in forming the sum we take account of the 

 fact that sin J (n7r/36) does not change when n is replaced by 36 M. For example, 

 let F be equal to 1 at the points indicated in the table, and zero at other points. 

 Then the contribution to the terms containing any m of the two parallels given by n 

 and 36 n is either 0, 1, or 2, according as a 1 occurs on neither parallel (for the 

 particular m in question), on one, or on lx>th. This number 0, 1, or 2 is to be 

 multiplied by tin- value of cos(mir/3G) for the chosen m ; but the same value for the 

 cosine occurs at the meridian given by 72 m, and the same numerical value with the 

 opposite sign occurs at the meridians given by 36 m and 36 + m. We condense into 

 one term the contributions of the eight points given by n, 36 n, nt, 72 m, 36 m, 

 and take the ranges of m and n to be respectively to 17 and 1 to 18. Thus, as the 

 multiplier of cos (mw/36) sin* (nn/36), we have an integral number which necessarily 

 lies between 4 and 4, and may be zero, and we have transformed the sum into a 

 double sum of the form 



17 18 _ 



_ 4 T^/ / \ flltt j ii'Tf 



2 2 F'(/i,m)cos sin*;-, 



m=0 = I OO OO 



where F' is the number in question. The most troublesome part of the process is the 

 determination of F'. When F' has been found it is very easy to form the sum of 

 such a series as that written immediately above by summing first with respect to m 

 and then with respect to n. When we are dealing with tessera! harmonics of the 

 second rank, we can thus condense into one term the contributions of 16 points of 

 the table, and, when the tesseral harmonic is of the third rank, those of 24 points. 

 Much labour is saved by going through this process, troublesome though it is, and 

 much greater accuracy can be secured, localise in the multiplication of cos (mtrf'SG) 

 by F', when F' is, say, 5 or 6, and the value of the cosine to any chosen number 

 of decimal places is used, it is easier to correct the figure in the last place than it is 

 when the same cosine occurs five or six times in a long column of figures which have 

 to be added together. 



60. By the use of this method I computed the values of the coefficients p, &c., for 

 the function F (0, <j>) which is given by the -1's in the table of 56, the 1's being 

 replaced by zeros. Up to the stage of summation with respect to m, inclusive, I kept 

 four decimal figures. Of the terms of the type 



I then kept two decimal figures, formed the sums with respect to n, and multiplied 

 them by the corresponding numbers placed in brackets in the third column of the 

 table in 58. This process gave the coefficients in the second column of the annexed 

 table. The integral parts only were retained. I computed the values of the 

 coefficients p, <fec., in the same way for the function given by the 1's in the table of 

 56, the 1's being replaced by zero. This process gave the coefficients in the third 



2 H 2 



