680 Dr. J. R. Wilton on Figures of 



so that the above equation becomes 



r\l + k 2 x 2 ) = a+ t0*% a sn [r(l + £V)*]» R, n 



s—l n—0 



= a+ | [r 2 {l + kW)]i*-R^i a sn 6'. 

 Whence, by Lagrange's expansion, we obtain 



FWl + ^)]=F(a)+J i ^ ! ^ r {[| o a^| i ^]"FXa)}, j 

 so that equations (B) become 



unless ?i=2, in which case the term (1 + P)(K — K) must 

 be added to the left-hand side o£ the equation. 



These equations must be satisfied for all values of 0, that 

 is to say the coefficients of the various powers of 6 on the 

 two sides of an equation are to be equated ; but it must be 

 remembered that K is a function of 6 which is determined 

 by the equation for which n = 2. Further, since the middle 

 point of the axis has been taken as origin, the value of r 

 when x = l is equal to its value when %= — 1; and it will be 

 found on substitution that the coefficients of all the odd 

 harmonics must be zero. We have thus proved the sym- 

 metry of the pear about a plane perpendicular to the axis of 

 revolution. 



We may assume that the equation of the pear is, using a 

 slightly different and rather more convenient notation, 



r 2 (l + kV) =--a + (a 10 + a n r 2 F 2 -f ai^ 4 P 4 ) 



+ <9 2 (a 20 + a 22 r 4 P 4 + a 23 r 6 F 6 ) 



+ 6* (a 30 + a 32 r 4 P 4 -f a 33 r 6 P 6 + « 31 r 8 P 8 ) 



+ &c, (12) 



where use has been made of the fact that harmonics of order 

 higher than 2s + 2 cannot occur in the coefficient of 9 s . 

 This fact may readily be proved ; its truth will appear in the 

 course of the work of determining the coefficients. We 

 have also determined (except its scale of measurement) by 

 taking it proportional to the coefficient of ? ,2 P 2 . 

 Equations (B) may now be written 



(1 + F)(K - K) + Ka n = i * P 2 log [r 2 (l 4 ifeV) ] dx, 



Jo 



