the Conservation of Force, 119 



and R, it therefore represents the sum of the intensities of the 

 forces which act at all distances between R and r. Calling 

 the forces which tend to move the point m . . . the tensions 



f R 

 . . ., then the quantity I (frdr would be the sum of the ten- 

 sions between the distances R and r." (Taylor's Scientific 

 Memoirs, 1853, p. 122.) 



It is somewhat surprising that a mathematician of Dr. Helm- 

 holtz's pretensions should have allowed himself to speak of a 

 " surface ... as the sum of [an] infinite number of ordinates," 

 that is, of an infinite number of geometrical lines ! The sum 

 of an infinite number of lines can but be a line, infinite it may 

 be in length, but not a surface. 



Dr. Helmholtz's surface is not the sum of an infinite number 

 of ordinates, but Ihe sum of the products of an infinite number 

 of ordinates each multiplied by the distance between itself and 

 its next consecutive ; in other words, the surface is the sum of 

 the products of the ordinates by the distances between them. 



Having misconceived the import of the surface which he 

 has thus introduced to our notice, we can hardly be surprised 

 that Dr. Helmholtz should have mistaken the character of the 

 integral of which that surface is a correct representative. 



In the differentiations which occur in this part of Dr. Helm- 

 holtz's memoir the time is the " primitive variable," a fact 

 which he himself expressly recognizes (ubi supra, p. 120). 



It follows that the quantities which he has thought proper 

 to write d(q 2 ) and dr would be more fully and precisely 



m M „„, * 



expressed by the symbols 7 J dt and -=- dt, and that his 



equation 



J md(q 2 ) = — fair 



is in effect identical with the following, viz. 



\m d Mldt=-$%dt, 

 2 dt T dt ' 



and that his equation 



f R 

 i mQ 2 — \mq 2 = — 1 fair 



Jr 



is nothing more or less than the following, viz. 



iQ 2 — ±mg 2 = — \ <l>£dt, 



where T and t are the times corresponding to the distances R 

 and r. It follows, therefore, that what Dr. Helmholtz would 

 have us accept as " the sum of the tensions [acting] between 



