166 



true for closed currents, cannot be inconsistent with known pheno- 

 mena, and may possibly be the symbolic representative of a real law 

 of nature. Such a law was suspected by Faraday from the first, 

 although, for want of direct experimental evidence, he abandoned his 

 first conjecture of the existence of a new state or condition of matter. 

 As, however, we have now shown that this state, as described by 

 him (Exp. Res. (60.)), has at least a mathematical significance, we 

 shall use it in mathematical investigations, and we shall call the 

 three functions a , [3 , y , the electrotonic functions (see Faraday's 

 Exp. Res. 60. 231. 242. 1114. 1661. 1729. 3172. 3269.). 



That these functions are otherwise important may be shown from 

 the fact, that we can express the potential of any closed current by 

 the integral 



J(« 2 «o^+^o|+^yo|)^ 



and generally that of any system of currents in a conducting mass 

 by the integral 



(a « 2 + fi b 2 + y c 2 ) dx dy dz. 



The method of employing these functions is exemplified in the 

 case of a hollow conducting sphere revolving in a uniform magnetic 

 field (see Faraday's Exp. Res. (160.)), and in that of a closed wire 

 in the neighbourhood of another in which a variable current is kept 

 up, and several general theorems relating to these functions are 

 proved. 



February 25, 1856. 



A paper was read, " On a direct method of estimating Velocities, 

 Accelerations, and all similar magnitudes with respect to Axes move- 

 able in any manner in Space, with applications." By Mr. Hayward, 

 of St. John's College. 



The frequent recurrence, in many different investigations of kine- 

 matics and dynamics, of exactly corresponding equations, suggests 

 the inquiry whether they do not result from some common principle, 

 from which they may be deduced once for all. An investigation 

 based on this idea forms the first part of this paper, and the result 

 is the method mentioned in the title. 



This calculus shows how the variations of any magnitude, capable 

 of representation by a straight line of definite length in a definite 

 direction, and subject to the parallelogrammic law of combination, 

 may be simply and directly determined relatively to any axes what- 

 ever. If such a magnitude (u) be estimated in a given direction, its 

 intensity in that direction will be represented by the projection on 

 it of the line which represents u. If this given direction be not 

 fixed, but move according to a given law, the projection of u upon it 

 will change by the alteration of its inclination to the direction of u ; 

 and the rate of that change is easily calculated, whence an expres- 



