macfarlane: algebra of physics 369 



doubt about the importance of the subject in the then state of 

 electrical science. 



Application of hyperbolic analysis to the discharge of a condenser. 

 Trans. Amer. Inst. Elec. Engineers, 14: 163-174. The inves- 

 tigation of the discharge of a condenser leads to a differential 

 equation, the solution of which depends on the solution of a 

 quadratic equation. The author proceeded on the following 

 theory of the quadratic equation. So far as line algebra is con- 

 cerned the roots of a quadratic equation with real coefficients 

 are either both real, or else conjugate complexes, the complex 

 roots being scalar in both terms. But for plane algebra the roots 

 of such an equation are either two conjugate hyperbolic roots or 

 else two conjugate circular roots; in both cases the roots are planar. 

 What is new in this theory is the treatment of the real roots as 

 conjugate hyperbolic roots. The truth of this principle was made 

 evident by the application to the discharge of a condenser. 



Sur la resolution de V equation du troisieme degre. Association 

 francaise pour l'avancement des Sciences. 1897. In this paper 

 the rules of plane algebra were applied to Cardan's solution of 

 the cubic equation x 3 + qx — r = o; viz. 



3 



\r I q 3 r 2 ) ' \r \ q 



x = l2' + \27+4J +I2-V27+4} 



When the quantity under the radical sign is negative, that is 

 in the irreducible case, the binomial expresses a circular complex 

 quantity; and when the quantity under the radical sign is posi- 

 tive, the binomial expresses a hyperbolic complex quantity. The 



q 3 

 hyperbolic solution has two cases ; if ^ is negative, the hyperbolic 



vector belongs to the primary hyperbola; if that quantity is 

 positive, the vector belongs to the conjugate hyperbola. In 

 every case the values of x were deduced by plane algebra, circular 

 or hyperbolic. 



Differentiation in space-analysis. Read before the American 

 Mathematical Society in 1895. Science 1 : 302. I stated that 

 there were two distinct kinds of differentiation, and that only one 

 of these was treated of in works on quaternions or vector-analysis. 



