116 



quantity, so as, for instance, to cliange any term k. —^ inta 



k "^'/ > and of course <p.x or —^^ into unity. Be" will satis- 

 fy the given equation ; B being any arbitrary constant, and c being 

 the well known transcendental 2,718, &c. whose hyperbolic loga- 

 rithm is unity. For, substitute this quantity for (p.x in the function 



F; any term of it, as ^. -^^, will become A;. —j^= A;Bc«^«''; 

 and the entire function V will therefore become Ec « ^ F„ ; Va. 



being what V becomes when for -j— we write «, and being 

 consequently, by the supposition, equal to nothing. This value of 



(p.x will therefore render equal to nothing F, = Be "^ V^ ; and 

 will consequently satisfy the given equation ; and it will in like 



manner be satisfied by Bc,"^ ^ JBc-^, &c. if «, «, &c. be also roots 



of the equation, V ^^ o, and B, B, &c. be other arbitrary con- 

 stants. It appears then that the quantity ^.x should generally 

 consist of n terms of the form Bc^, and is therefore, as I said 

 above, in its nature exponential. If any roots of the equation, 



V = o, should be impossible, the impossibility may be removed by the 

 following artifice. Let, for example, € + ^ V — 1 be roots of the 

 aforesaid equation. The corresponding parts of (p.x will be 



(? + \^—i)x . (S— x^— l)x 



F c + F c Feign F and F to be respec- 



tively equal to ~ o , and ■ ~ , B and B being ne»v 



