On the Solution of certain Pi'oblems. 29 



where 



so that the number of disposable and independent quantities 

 (S) contained in the expression for 7/ can in no case exceed 

 11 ; neither, after the elimination of the nine ^'s, can it e:^ceed 

 13, as is seen on referring to the last of the above equations. 

 The elimination of those quantities does not, however, dimi- 

 nish the number of disposable quantities (H) except when m 

 is greater than 13. 

 If then 



h,=KlS'-\- Kl< S" + ^i'" S'" + is:,iv s^y + B^ 



h^=Ki' S'<+Ki" S'" + K^^ Siv + 5^ 



where By, . , . , B^ are functions of the n — 5 quantities S\ 

 S", . »'""'', we see, by what precedes, that nine of the quan- 

 tities ^ may be so determined as to enable us to decompose 

 gYn = into 



h,^ + h^^=o, . (A.) and hs^ + /i4^ = 0, . (B.) 

 where /i,, .,.,7^4 have the forms last above given, and the 

 n — 5 quantities S^, S^*, . , H^"~^^ are undetermined, and per- 

 fectly at our disposal; at least when n is not greater than 13, 

 and when 7i exceeds 13, we have eight of them undetermined 

 and disposable. But it will be seen below that, for our pre- 

 sent purpose, this last case does not require consideration. 

 E', S", . E% have as yet no other conditions than (A.) and 

 (B.) to satisfy. 



Depress (A.) and (B.) to linear equations, and eliminate 

 B'", S'^, from gY',, by their means. Then, on referring to my 

 paper at pages 190-191 of the last volume but one of this 

 work*, it will be seen that, without determining a', S", it will 

 be possible to reduce the resulting equation to the form 



or, h,'^ + h,^ = 0; (C.) 



and also that B, and Bg will not give the illusory results 

 which (under a diderent notation) I have beforef pointed out, 

 provided the number of disposable quantities H^, . , . , S^"~^ 

 be more than three in number; this gives the condition 



7J — 5 >3, or ?i> 8 (y.')t 



With the aid of (C.) reduced to a linear form, eliminate 3' 

 or S" from ^Y'n = 0, and solve the resulting ccjuation. 



• Phil. Mag. S. 3. vol. xxviii. f Ibid. pp. 190,1 91. ;]9f). 



I This corresponds to the equation (y.) of p. 191 of Phil. Mag. S. 3. 

 vol. xxviii. 



