28 
Proceedings of Royal Society of Edinhurgli. 
A = (2^ + + q^-\- - ’"llP-cf + '^p^f , 
= - 2(p2 _ (f){p^ - ^2^ 4- {^(P + ^2^2 ^ 
= 0 + (2a6)2; 
A = 3 a 8 + 3 &S + 3 p 8 + 3 g§ - 4a6^2 _ 450^2 _ 4^e^2 „ 4 ^g^ 2 + i 2 ^Y + 12$6p2 \ 
- 6(2^6^ ~ I 
+ ia'^lP'p^ + 12a^p^(f + ib^odf + Ylh^fcf - ^p^cdf > 
- d>p%\^ - ^(flP-p^^ - '^(fa^p‘^ I 
- 48a2Z;2^ V ; ) 
(j = + V^ -p^ -q^ . 
A complicated result like this, dependent upon lengthy calcula- 
tions, is more of a hindrance than a help, unless it can he perfectly 
relied upon. 
One test as to its accuracy is suggested by the constitution of 
the original equations (a) and (/?), which on examination will be 
seen to be symmetrical with respect to the interchange 
La ^ 
\ p q\ 
\ X y ) . 
Now a glance at O, V, <E>, 0, 'k, p, A, A, o- will be sufficient to 
show that they are all symmetrical with respect to the inter- 
change 
(a 
I P s 
and this being ascertained it is at once manifest that the resultant 
is, as it ought to be, symmetrical with respect to the same inter- 
change as the original equations. 
Next, there is the test of its agreement with the results otherwise 
obtainable for special cases, — a test which if satisfied is additionally 
instructive in that it makes evident the mode in which the degene- 
rate forms of the equations arise. 
Taking in the first place the case where p~q = 0. We have 
O = (a2 - h^f = r = l<^, X = - 5^)2 A, 
0 = (a4 - h^) {a? - 62) = 1^, A = (a2 - 1 - h^)\ 
p — a^-\-l)^ = (T, 
and the equation becomes 
