140 MR. JAMES COCKLE; RESEARCHES 
to equations which present as much or more difficulty than 
the given one, then the solution is, not absolutely, but 
conditionally or relatively, unattainable. The latter con- 
clusion may afford probable evidence that the solution is 
impossible ; but the impossibility cannot be said to be 
demonstrated. 
§ 27. 
Let 
vo” + iw” + Pa” + By” +2" =f"(i), 
f° @®F/"@H“*)=7,(u”), and 
PB) FP) =r(w"). 
Then 
7 (Ww) + 72(u”) = 230 — So™w", 
— 5ry(u") x 7,(w") = 0(w") + K 
where 6(w”) is the same function of w” that @ or O(u) is of 
u, and K is known, and, consequently, 7 is known when 
@ is known. The function 6(w”) is known whenever 6(w) 
or @ is known. 
§ 28. 
Let 
Le) yi (2*) +f"(#) t'() = L, 
POLO FLO LE) =S, 
then 
I aL AL =4> 7" ie Sv"w", 
and 
=3{(+I~-@4+9)}. 
But 
PPS LPOPOP+RPOLOPHPOCE LOY 
+1 fr@ Pe oy 
os Mena "|\POPLPOAPOLOLOLES 
Sut ) 7y(w) + T9(w”) To(w”) 
a re = is a function of tr. Consequently I and J are 
known when 7, which depends upon 0, is known. 
