182 Proceedings of Royal Society of Edinburgh. [sess. 
NX0* = * + (0,1)0^ + (0, 2)0* 2 + 
NXj&B = (1,0)0* * + (1, 2)0* 2 + 
XX 2 0* = (2,0)0* + (2,1)0^ + * + 
. + (O,p)0*p ) 
. +(l,p)0*p 
. +(2 ,p)dx p 
XX p 0* = (y>,O)0* + (^,1)0^ + (p, 2)0* 2 + . . . . + * 
where (0,0)= -(1,0) and generally (a, (3) + (/3,a) = 0. This form 
of his own he frankly characterises as “elegant and completely 
symmetrical”; hut the same description would apply equally 
appropriately to the solution which he gives. Unfortunately, the 
method by which the latter was obtained is not indicated, and 
we can only hazard a guess in regard to it. The balance of 
probability would seem to be in favour of the method of devising 
a set of multipliers which, when applied to the given equations, 
would after the performance of addition bring about the elimina- 
tion of all the unknowns except one. In the case of four equations 
this would not be at all difficult. For example, if we wish to 
eliminate * 2 , * 3 , * 4 from the equations 
. ax 2 + bx 3 + c* 4 = £ 4 
- ax x . + dx 3 + e* 4 = £ 2 
- bx x - dx 2 . + /* 4 = £ 3 
C* 4 — 6*2 fx 3 . = $ 4 . , ) 
the multipliers are readily seen to be 
0, /, - e, d, 
so that after multiplication and addition there results 
( — af+ be - cd)x x = /£ 2 - e£ 3 + dx 4 . 
Similarly by using the multipliers - /, 0, c, - b we find 
( - af+ be - cd)x 2 - -/f 4 + c£ 3 - b £ 4 ; 
and the other two are 
(-af+be-cd)x 3 = e^-c^ + a^, 
( - af+ be - cd)x± B -d£ x + bg 2 - ag 3 . 
Jacobi’s corresponding result is to the effect that the numerators 
of the values of the four unknowns are 
N0*{ * + (2,3)X 1 + (3,1)X 2 + (1,2)X 3 }, 
X0*{(3,2)X + * + (0,3)X 2 + (2,0)X S }, 
N0*{(1,3)X + (3,0)X 1 + * + (OJJXg}, 
N0*{(2,1)X +. (0,2)X 4 + (1,0)X 2 + * }, 
