250 [Recess, 



In the third part of the memoir, the volume of any pedal of the 

 ellipsoid 



is expressed by means of the three first partial differential coefficients 

 of the symmetrical integral 



If P denote the volume of the pedal whose origin has the co- 

 ordinates x, y, e, the expression in question is 



where 



3M 1 = 

 3M= 



y a and being abbreviations for s?+y* + a? and 1 + 2 + 3 , respec- 

 tively. 



The memoir concludes with the expression of the volume P by 

 means of ordinary elliptic functions, and the consideration of the 

 special cases when the primitive is an ellipsoid of rotation. The ex- 

 pression in question may be readily obtained on observing that the 

 integral V is reducible to the form 



where the amplitude 6 and modulus k of the elliptic function P of 

 the first kind are determined by the relations 



By the introduction of elliptic functions, however, the great advan- 

 tages of symmetry are necessarily lost ; and in investigating the pro- 

 perties of pedal-volumes, the above symmetrical expressions will in 

 general be preferred. An opportunity thus presents itself, however, 

 of verifying an expression for the volume of the central pedal, the 

 only one hitherto calculated, which was first given in 1844 by Prof. 

 Tortolini in vol. xxxi. of Crelle's Journal. 



