Crofton — Applications of the Method of Operative Symbols. 607 
= exp. [ + ^) V - X- c] f{x^) OS 
.-. Dr f(a:») = .-i^f ^^y^i"^/(a;n). (16) 
Hence the question is reduced to finding the value of 
where is a constant as regards x, 
7. Por example, let 
w = 2, c={'^^)-^^; (5), 
Hence = 
1 
See 'Williamson's Diff. Calc, p. 448. 
This formula might be found independently of (15) : thus 
J^^fi'^') =/ + 2^ ^ + ^2) = ^^'"''''^^ ; Pitting ^ = J ; 
= [ (2^0'- + r (r - 1) (2iC)'-^C + ^^^^F^ i'^^Cr'C + • • .]/(^'), 
giving the same result. 
8. So little do we understand the methods of dealing with the 
symbols we are considering, that it may be useful to show how the 
formula (15) may be arrived at by another process. 
