548 Mr. G. B. Jerrard's Beflections on the Resolution 



tions, and to show that those values of p^, pci^ which satisfy' 

 them, will necessarily satisfy every other pair belonging to the 

 same system. 



In effect, if introducing an indeterminate multiplier \ we 

 unite 



with <|) = fX^t/^ + p,^j/^ + fM^IJy + f^sT/S -h ^,?/„ 



on dividing the result by ft^ + A, and designating 



we shall find 



Now in order that ^ may be different from 



v-o. {y<. ^y^+ Vy + i/s + y^'> 



X must admit of being determined so as to satisfy at least one 

 equation of the form ju.^^ + X = 0, without causing jw,^ + A to 

 vanish. If, therefore, we reflect that the system of equations 

 on which $ depends will remain unaltered if, while we sub- 

 stitute another imaginary root i' instead of i, we make certain 

 substitutions among ^^, j/, j/g, t/. ; and that consequently <I> 

 may be deduced from 



y„« being the same function of »' and A' as v„ is of < and A, 

 but y„' being a different root from y^: we shall readily per- 

 ceive that the coefficient of any one indifferently of the four 

 roots 2/^, «/y, 2/3, 2/6 niay be equated to zero, when the coeffi- 

 cient of 2/a is equal to I . 



Accordingly let us suppose that 



and, on expressing j/^, 2/^, . . 2/s in terms of / and ii, there will 

 arise 



^ . . , . 1 



This expression for must m vanishing assume the 



form Ot + Ou, For — , which is not independent of A'gj A'g, 



