96 Mr. J. J. Guest on the Strength oj 



or 



or to the first order, 



y 2 {l — 0i— «i) — ~ 2 </>i = 0; .... (6a) 



hence ^ 2 is a small quantity. 

 Substituting in (3), 



2 i 2 1 



;V + S 2 — 1, 



which combined with (5) gives us as values x 2 — and z 2 —\. 

 (The values x 2 = l and z 2 = correspond to K.) 



Also it is evident that l — z 2 = ^ is of the second order. 



[For 



^ 2 + y 2 2 +- 2 2 = ^ 2 2 + y 2 2 +(l-X) 2 



= l-2X+(.r 2 2 + ?/ 2 2 + X' 2 ); 

 .*. X is of second order.] 

 .". (4 a) becomes # 2 — 1 = 1 + a 1 , 

 and (6a) becomes y 2 —cfi l = 0; 



hence # 2 = 2# 1 + a, -^ 



y2 = <j> 1} > to first order. 



'z 2 = l, J 



Let the terms of the second order in x 2 , y 2 , z 2 be \, fi, v 

 respectively; then from (4) 



(20 1 + « 1 + X)(l + 1 )+fc»+(l + v)(l-0i-^) 



Hence 



2 2 



X + v= -2^-2^0- ^-|Ui 2 . . . (7) 



Substituting in (3) , 



40! 2 4- 40!**! + «i 2 + <f>i* + 1 + 2j/= 1 ; 



.-. v=-(20 1 2 + 20 1 . 1 +9- 2 + ^); 



