116 
Proceedings of the Royal Society 
and, of course, ^ ^ 
Also, <1p= — 3 = Tp<1 Up+ <1 Tp . Up = Tp<3 Up — 1, 
••• <3Up=-^ (7) 
111. By the help of the above results, of which (0) is especially 
useful (though obvious on other grounds), and (4) and (7) very 
remarkable, we may easily find the effect of <1 upon more complex 
functions. 
Thus, <1 Sap= - <1 (aa? + &c.)= - a (8) 
<lYap= — <lVpa= — <I (pa — Sap) = 3a — a = 2a (9) 
Yap 2a 3pYap 2ap^+3pYap af ~ opSap _ . 
Hence 0^^ = Tp- 
TT G! S ^ _p^Sa8p — 3SapSpSp _ Sa8p 3SapSpSp 
Hence . dp<l 
= ( 11 ) 
This is the principal transformation alluded to in the title of this 
note. By (6) it can he put in the sometimes more convenient 
form 
S.Sp<]^ = 8S.a<il (12) 
And it is worthy of remark that, as may easily be seen, S may be 
put for Y in the left hand member of the equation. 
We have also 
<1 Y. /?py = < {^Syp - pS/3y + yS/?p} = - y/? + 3Sy8y - (By = S/3y. (13) 
Hence, if <}> be any linear and vector function of the form 
0p = a + 2 V. /Spy + mp, (14) 
then <1 </>p= — 3m= scalar (14)^ 
Hence, an integral of 
== scalar constant, is cr- = <f)p.. (15) 
If the constant value of <l <r contain a vector part, there will be 
