80 
PROFESSOR DONKIN ON THE 
(see equations (6.) and (9.), putting a { for p in the latter) exactly in the same way, 
it is plain that the result may be deduced from (18.) by interchanging x and y, and 
changing the sign of b ; thus 
dx k ' 
djk 
' da j 
Lastly, from the equations 
dX a dX b , , . 
■W =a “ *7 = -^ < see < 12 -) and 
we should find in a similar manner 
da. j dx/c . 
dy k dbj 
and from the analogous equations (the existence of which is obvious) 
dY b dY b 
dyi Xi ’ dbi a% 
we should obtain 
d^i. 
dxd 
dyk 
dbj - 
Collecting these results, and changing the indices, we have the system 
dxi 
dcij 
dbj 
dyl 
dxi da,j 
dy t 
dbj 
dyi dbj 
dcij 
daj 
(19.) 
dxi dbj dxi 
in each of which equations the first member refers to Hyp. I., and the second to 
Hyp. II. (art. 5.) ; and it is to be remembered that there is no relation between the 
indices of the variables and those of the constants, so that the case of i—j has no 
peculiarity*. 
8. Let ci, A be symbols denoting two distinct sets of arbitrary and independent 
variations attributed to the 2 n constants ; then the equations 
dX 
dX 
= b. 
dxi dcii 
give ^X=2 i (»/M+«i^i) ; 
and if the operation A be performed on each side, we have 
A^X=2j(A y$Xi-\- &a$bi) 
+%(yAlx i +aAlb i ). 
* It is remarkable that each of the equations (19.) is also true on a different and separate hypothesis, as is 
apparent on inspection of the four different sets of equations, 
dX _ dX b dY dY b 
_ dX b _ _ 
, — yu —j — ~yi> -j — x i> 
dxi dxi dyt 
—=b- 
dai 1 
dX b _ 
db; 
dY , 
dai 
-= —Xi 
dyi 
dbi 
(see the preceding articles). 
