294 On a Question arising from a recent Mathematical Controversy. 



example, by means of the known expressions for the area or 

 content of a triangle, ellipse, pyramid, ellipsoid, or cone, this 

 theorem enables us by differentiation and algebraical processes 

 alone to obtain the parameters which define the centres of gra- 

 vity, moments of inertia, principal axes, &c. of such figures. 



I must add an important observation, viz. that the theorem 

 remains true when one of the defining equations (supposing 

 there to be more than one), instead of being the most general of 

 a certain degree and kind, is affected with arbitrary numerical 

 coefficients (zeros or others), provided only that it be homogeneous 

 in the variables. Again, the theorem continues to hold when 

 the original density, instead of being a homogeneous function of 

 the variables, is such function multiplied by any Covariant of the 

 defining equations taken separately or in groups — using the word 

 covariant in its most extended sense, so as to comprehend frac- 

 tional and irrational as well as integral forms, — the only effect 

 of the introduction of such new factor into the density being to 

 modify the form of the differential operators. There are certain 

 very special cases, to which it is not necessary to allude here in 

 detail, in which the theorem becomes illusory : such will be the 

 case, ex. gr., for a plane area when the given density is a homo- 

 geneous function in the variables of the negative degree 3, and 

 for a solid content when that density is of the negative degree 4*. 



K, Woolwich Common, 

 September 1863. 



XLI. On a Question arising from a recent Mathematical Contro- 

 versy. By G. B. Jerrard, Esq.f 



HAVING for some time turned my thoughts away from 

 mathematical subjects, I did not discover till lately that 

 the present Number (that for September) of the Philosophical 

 Magazine contains a passage involving an objection the relevancy 

 of which I ought not for a moment to let pass unquestioned. 



number of functions and any number of variables. Dr. Arm hold, in the 

 last Number of Crelle's Journal, states erroneously that these equations 

 were given by me for binary functions only, and subsequently generalized 

 by Cayley and Clebsch. 



* A similar method applied to extents (as curves, surfaces, &c.) gives 

 rise to curious theorems. Thus I find that the mass of a plane curve 

 affected with a density varying at each point as the square of the cosine of 

 the inclination of the curve to a fixed line, is a differential derivative of the 

 length of the curve- So, again, the moment of inertia of a curve in respect 

 to any axis perpendicular to its plane, is a differential derivative of its mo- 

 ment in respect to an arbitrary line in its plane, 



t Communicated by the Author. 



