537] 
SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
515 
( cloc\ 
= j is therefore obtained from that of x by differentiating in 
regard to t alone, as if a, e, c, m were constants: viz. x will be a given function of 
t, a, e, c, m; 8x' then denotes the variation of x in so far as it arises from the 
. . . . dCl 
variation of the constants, viz. the equation 8x' = ^ clt means 
dx' da dx de dx de dx d'or dii 
— [. — 1_ - 1 = 
da dt de dt dc dt dm dt dx 
There are the like equations in regard to y, y', viz. in all, four equations which are 
da de dc 
dt ’ dt ’ dt ’ dt 
linear in regard to ~^^^; and which serve to determine these quantities in 
terms of 
dii dii 
dx ’ dy 
Now considering the x, y as expressed in terms of a, e, c, m, t, then il becomes 
a function of these quantities ; the differential coefficients , &c., being connected 
with the original differential coefficients ~ by the equations 
di1 _ dil dx dil dy 
da dx da dy da ’ 
dfl _ di1 dx dil dy 
de dx de dy de ’ 
&c. 
As there are four equations, can be expressed in an infinity of ways in 
terms of 
dil dil dil dil 
da 
^da’ de ’ dc * dm’ an< ^ considering ^, &c., as given in terms of 
da 
dii dil 
dx dy ’ 
dii dii dii dil 
we can in an infinity of ways express -r-, &c., as linear functions of 7 , 7 , 7 , , . 
dt da de dc dm 
But there is one form (obtained by combining the equations in a particular manner) 
wherein the coefficients of the last-mentioned quantities are functions of a, e, c, m 
without t; and this is the form actually employed for the expression of } 
. „ dil dil dil dil . . . , . , 
m terms or ^^, m the method wherein these quantities are made use of. 
I remark upon the present question, that the answer ought to be in substance 
perfectly familiar to every student in Physical Astronomy; and that a student ought 
to be able to present it in a clear and logical form: the question being in fact 
intended as a test of ability in this respect. 
65—2