316 Mr. W. P. Boynton on the 



nating E, F, G, H, and noting that we may write 



k 1 = a-\-bi J k i = c-\-di,~\ 

 k 2 = a — bi, k i =c-'di ) J 



we obtain the four relations 



(12) 



aA 1 + bB 1 =A 2j — 6A 1 + «B 1 =B 2 A ,^ 



cG 1 + dD 1 =C 2 , -rfd + cD^Da, J 



The initial conditions that when t = 



rwi-f urnon / A 



Q 1 ^Y 1 K 1 =\,K 1 , §1=0. Q 2 =0, §=0, 



-«A 1 +/9B 1 - 7 C 1 + 8D 1 =<n (U) 



-aA 2 +/3B 2 - 7 C 2 + SD 2 =0,J 



give the four equations 

 A 1 + C 1 =Y K 1 , 



A 2 + O 2 = 0, 



which suffice with equations (13) to determine all the eight 

 constants. 



If from equations (14) we eliminate A 2 , B 2 , C 2 , D 2 by 

 equations (13) we have four equations in A 1? B 1? C 1? D^ 



A, + G 2 =V K 1? ^ 



aA l + bB 1 + cG l + dD 1 =0, I 



-aA 1 + /3B l - 7 C 1 + SD 1 =0, f v } 



(«a + i S6)A 1 + (a&-/Sa)B 1 + (70 + 8^)0! + (7^-Sc)Di=0/ 

 whose determinant is 



A =/38(a 2 + b 2 + c 2 + d 2 ) -bd(o? + /S 2 + 7 2 + S 2 ) + 2*ybd- u 2l38ac, 

 and their solution 

 A = YpK^^Cc 2 + d*) -bd(y 2 + S 2 ) + {ab-Pa) (yd + Sc) ] 



A 



B,= 



V K 1 [^(c 2 + d 2 ) -f 0<% 2 + S 2 ) - («g + /3b) {yd + 8c) ] 



A 



n y K 1 [^S(« 2 + 5 2 )-^( a 2 + y 8 2 )+( a 6 + /3«)( 7 ^-Sc)] ^ (1H) 



n _ VoKi [/3 7 (a 2 + 6 2 ) + bc{* 2 -f /3 2 ) - (*b + /3a) ( yc + Bd) ] 

 l>i- 5 , J 



Or, if we disregard the squares of the small quantities a ; y, 

 b, d (- is of the order of -~, &c>), 



