PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 17 



' rrr, . d V 



\// = III (u x + v y + w z) ~{r>af) (24). 



17. To obtain the part of the displacement £ due to the initial velocity of dilatation, 

 we have only to differentiate \|/ with respect to x, and this will be effected by differentiating 

 u , v , w under the integral signs, as was shewn in Art. 5. Treating the resulting expres- 

 sion by integration by parts, as before, and putting I, m, n for the direction-cosines of the 

 radius vector drawn to the point to which the accents refer, and £, for the part of ? due to F, 

 we get 



£i = v. //{('«• + i""o+ nw o)„ ~ ( lu o + mv o + nw ) } dy dz 



4 wO t J J 



t rrrf d x d y d ss\ 



+ 7~ /// \ u °7--3 + "»r^ + M, ori \dV(r>at). 

 47r JJJ \ das r 3 dwr door 1 } 



Let q 9 be the initial velocity revolved along the radius vector, so that q =*lu + mv + nw , 

 and let (q )at be the value of q at a distance a t from O ; then 



ff{(lu + mv + nw a ) it - (lu + mv + «w ),} dy dss = ffl (q^ dS = effffl (q )at d <r> 



d x d y d % u — 3lq 



and u — - + v — - + w — - = . 



°dxr % dxr 3 dxr* r 



Substituting in the expression for f u we get finally 



^ = l r If l ^)a t da + ~fff(u Q -3lq ) d ^(r>at) (25), 



18. Let us now form the part of f which depends on the initial rotations and angular 

 velocities, and which may be denoted by £ 2 . The theorem of Art. 14 allows us to omit for 

 the present the part due to the initial rotations, which may be supplied in the end. Let 

 <*>o'> w o " ■> w o" be the initial angular velocities. Then £ 2 is given in terms of w" and •ar'" by 

 the first of equations (16), and -ar", ■ar"' are given in terms of w ", w '" by the formula (7), in 

 which however b must be put for a. We thus get 



The integrations in this expression are to be understood as in Art. 15, and w ", w '" are 

 supposed to have the values which belong to the point Q, but PQ is now equal to bt instead 

 of at. The quintuple integral may be transformed into a triple integral just as before. We 

 get in the first place 



i-Mffhy r ff"-£-fif« ir ff"-£)- ■ ■ ■ ™- 



The double integration in this expression refers to all angular space, considered as extend- 

 ing round Q; x, y, % are the co-ordinates, measured from 0, of a point P situated at a 

 distance bt from Q, and r = OP. If dS - {btfda, the expressions for the integrals 



ffxr~ 3 dS, ffyr~ 3 dS, ffzr~ 3 dS 

 Vol. IX. Part I. 3 



