of Edinhurgli, Session 1883 - 84 . 
585 
10. The theorems of the preceding four paragraphs, or even of 
§§ 6, 7, 9, afford a complete solution of the equation of condition. 
The process may be thus described. It depends upon the solution 
of equations of two types, A and B say ; of which 
(v, w, X, y, z) = 2(ti, V, IP, X, y) (A) 
and (v, 10 , X, y, z) = {v, to, x, y) + (w, y, z) + {w, x, y) . (B) 
are examples. The equation of condition, to begin with, is of the 
type A. We show that this can be made dependent upon an equa- 
tion of the type A but of lower order, and upon an equation of the 
type B. Next we show that the equation of type B depends on the 
solution of another of type B of lower order, and on the solution of 
two equations of type A. Thus by repeated partial solutions the 
desired result is obtained. 
A caution, however, is requsite in regard to the ultimate equations 
of the two types. In the case of type A the ultimate form may bo 
either (x, y, z) = x, y) (kz) , 
or {y, z) = 2{x, y) (A^) . 
The former has for solutions 
(1) z=2w,x:=2y, 1 
(2) z = 2w+\, x = \,y = 2 ■,] 
the latter has 2=2.r-l-l,2/ = l • 
In the case of type B the ultimate form may be either 
{x, y, z) = {x, y) + [y, z) + y (Bz) 
or {x,y)= X + y +l (B.^) 
The solutions of the former are 
(1) x=l,z=2y+l , -V 
( 2 ) x = y = z=2, 
(3) 2y + 1, = 1 : 3 
of the latter x = y = 2. 
11. The application of the theorem in § 8 effects a considerable 
simplification of this process. W^henever an equation of type B is 
reached, it enables us to dispense with the consideration of the last 
and most troublesome of the three partial solutions given in § 9, 
For, this solution is 
