Equivalent Resistance fyc. of Parallel Circuits. 271 



particular concretes of the order of angles and angular dis- 

 placements are treated as pure numbers. The present paper 

 may be regarded as an attempt to remove such difficulties by 

 taking as fundamental in our theory of dimensions the vector 

 instead of the cartesian unit length. By expressing the for- 

 mulae according to the conventions of vector algebra, we thus 

 assign dimensions (in the extended meaning of the term) to 

 all quantities which are physically recognized as concrete, the 

 only quantities having unity as their dimensional formula 

 being pure numbers and quantities of the nature of tensors, 

 that is, ratios between similar and similarly directed concretes. 

 Since, as pointed out by Prof. Lodge, the cause of the above 

 difficulties lies in the neglect of the directional relations of 

 quantities, these modified formulae approximate more closely 

 to what they are sometimes, and conveniently taken to be, 

 namely conventional expressions of the kinds of different 

 physical quantities. 



XXVIII. Equivalent Resistance, Self-induction, and Capacity 

 of Parallel Circuits with Harmonic Impressed Electromotive 

 Force. By Feederick Bedell, Ph.D., and Albert C. 

 Crehore, Ph.D.* 



IN a paper on " Forced Harmonic Oscillations of various 

 Periods," in the Philosophical Magazine for May 1886, 

 Lord Rayleigh derived analytically expressions for the equi- 

 valent resistance and self-induction of parallel circuits in terms 

 of the resistance and self-induction of each branch. The 

 object of this paper is to obtain by other methods similar 

 expressions for the equivalent resistance, self-induction, and 

 capacity of any number of parallel circuits in terms of the 

 resistance, self-induction, and capacity of each branch. From 

 these general results, particular expressions for the case of 

 circuits with resistance and capacity alone, and for the case 

 of circuits with resistance and self-induction alone, are readily 

 derived. The results in this latter case are identical with 

 those given by Lord Rayleigh in the paper referred to. 



Resistance is denoted by R, self-induction by L, capacity 

 by C ; and equivalent resistance, self-induction, and capacity 

 by R', L' ' , and C respectively. The currents (maximum 

 value) in the main line and branches are denoted by I m , and 

 by I 1( I 2 , &c 6 is the angle of advance or lag according to 

 whether it is positive or negative, co is the angular velocity, 

 i. e. 27r x frequency, and E the maximum value of the 

 impressed electromotive force. 



* Communicated by the Authors. 



us 



