Mr. VV. H. Walenn on the Unitates of Powers and Roots. 117 

 Then the preceding equation reads : — 



Mz=\V$}ogU+g8c+g f 8c ! (92) 



We can construct an equation of this form for each of the 

 three directions of coordinates. In doing so we put, for di- 

 stinction, the indices 1, 2, and 3 to each of the quantities occur- 

 ring in the equation, with the exception of \V, to which we ascribe 

 a common value for the three directions of the coordinates. If 

 we add together the three equations thus produced, and multiply 

 the sum by j^Nm, we shall get 



i^m8(h l +\ + ~h 3 ) = i NmU)S log (tl^tta) 



+ i Nm (^Sc, +tf l $(/ l +ff^c 2 +g' 2 8c' 2 +g 3 8c 3 +g' 3 8c r 3 ) . (93) 



But JNm(^, + h 2 + h 3 ) is nothing else but the mean ergal of the 

 entire system of points. If we now employ the letter U for the 

 ergal relative to a definite time of the entire system, we should, 

 properly, denote the mean ergal by U ; but as with a very great 

 number of points which have different phases in their motions 

 the value of the total ergal is always very nearly equal to its 

 mean value, we can omit the stroke over the U. The product 

 jiNmW is one third of the total mean vis viva; we therefore de- 

 signate it by IT, omitting here also the horizontal stroke over 

 the T. As regards the quantities g v g' 2 , g 3 , &c, with the present 

 signification of U and again omitting the stroke over its differen- 

 tial coefficients, the following equations are valid : — 



1 M - ^U 1 M — dV 1„ - d\J Q 

 2 Nm ^ = SV 2 Nm ^ 1 = SV 2 Nm ^=< &C - 

 If we insert these values in equation (93) and collect the terms 

 with the variations Sc v 8c\, 8c 2 , &c, under a sign of summation, 

 we get 



8U==gT81ogtt l u^+S^K&. . . . (9i) 



[To be continued.] 



XV.. On Unitation. — IV. The Unitates of Powers and Roots: 

 developments of these, with applications. By W. H. Walenn, 

 Mem. Phys. Soc* 



1. r | MAKING up unitation where it was leftf, treating that 



-i- which has been already put forward as introductory, 



and merely noticing that, if a number be expressed in terms of 



* Communicated bv the Author, 

 f Phil. Mag. S. 4. "vol. xlix. p. 316. 



