PHILOSOPHICAL TRANSACTIONS. 



407 



TOL. LXXXVIII.] 



its attraction in the direction ac, except by an infinite series. The two most con- 

 venient series I know, are the following : 



First series. Let - = tt, and let a = arc whose tang, is tt, b = a — tt, 



it* 



c== B _|. — p = c — y, &c. then the attraction in the direction ac = V 1 _ 



X (a + 



3 



IV 



. 3cw 4 ,3.5a; 6 x 



For the second series, let a = arc whose tang. = -, b = a , c = b -f- 



I 



(A- 



D == C — 

 3ciH 



BV 1 



+ 



5sr* ' 



3 . 5dv 6 



&c. then the attraction = arc 90° — ^ [(1 -4- ^ 2 ) x 



, &c.)] 



2 " 2.4 2.4.6 : 



It must be observed, that the first series fails when ?r is greater than unity, and 

 the 2d, when it is less ; but if h is taken equal to the least of the 2 lines ck and cb, 

 there is no case in which one or the other of them may not be used conveniently. 



By the help of these series, I computed the following table. 



Find in this table, with the argument -r at top, and the argument -?■ in the left 



QfC (lb 



hand column, the corresponding logarithm ; then add together this logarithm, the 

 logarithm of — t and the logarithm of -7- ; the sum is logarithm of the at- 

 traction. 



To compute from hence the attraction of the case on the ball, let the box dcba, 

 fig. 1, in which the ball plays, be divided into 2 parts, by a vertical section, per- 

 pendicular to the length of the case, and passing through the centre of the ball ; 

 and, in fig. 9, let the parallelopiped ABVEabde be one of these parts, abde being 

 the above-mentioned vertical section ; let x be the centre of the ball, and draw the 

 parallelogram finpmSx parallel to -&bdj), and xgrp parallel to j3b^w, and bisect $$ in c. 

 Now the dimensions of the box, on the inside, are b£ = 1.75; bd = 3.6; 

 B0 = 1.75 ; and (3a = 5 ; whence I find, that if xc and jar be taken as in the 2 

 upper lines of the following table, the attractions of the different parts are as 

 follows. 



