of the Law of Efficiency of an Electric Motor. 127 



ratio of the areas AFHD and GLCH, which represents the 

 efficiency, can therefore only become equal to unity when the 

 square KGHD becomes indefinitely small — that is, when 

 the motor runs so fast that its counter electromotive force e 

 differs from E by an indefinitely small quantity only. 



Further, it is clear that if our diagram is to be drawn to 

 represent any given efficiency (for example, an efficiency of 

 90 per cent.), then the point Gr must be taken so that area 

 GLCH = ft area AFHD ; or, G must be ft of the whole 

 distance along from B towards D. This involves that e shall 

 be equal to ft of E ; which expresses geometrically the law of 

 maximum efficiency. 



It is strange that even in many of the accepted text-books 

 this law is ignored or misunderstood. It is indeed frequent 

 to find Jacobi's law of maximum rate of working stated as the 

 law of efficiency. Yet as a mathematical expression the law 

 has been known for many years. It is implicitly contained 

 in more than one of the memoirs of Joule ; it is implied also 

 in more than one passage of the memoirs of Jacobi*; it exists 



* Jacobi seems very clearly to have understood that his law was a law 

 of maximum working, but not to have understood that it was not a law 

 of true economical efficiency. In one passage (Annates de CJiimie et de 

 Physique, t. xxxiv. (1852) p. 480) he says : — " Le travail mecanique maxi- 

 mum, ou plutot Veffet economique, n'est nullement complique avec ce que 

 M. Miiller appelle les circonstances specifiques des moteurs electromag- 

 netiques." Y et, though here there is apparently a confusion between the 

 two very different laws, in a preceding part of the very same memoir 

 Jacobi says (p. 466) : — " En divisant la quantite de travail par la depense 

 (de zinc), on obtient une expression tres-importante dans la mecanique 

 industrielle : c'est l'efi'et economique, ou ce que les Anglais appellent 

 duty." Here, again, is a singular confusion. The definition is perfect ; but 

 " etfet economique " is not the same thing as the maximum power. 

 Jacobi's law is not a law of maximum efficiency, but a law of maximum 

 power ; and that is where the error creeps in. It is significant, in suggesting 

 the cause of this remarkable conflict of ideas, that throughout this memoir 

 Jacobi speaks of work as being the product of force and velocity, not of 

 force and displacement. The same mistake — common enough amongst 

 continental writers — is to be found in the accounts of Jacobi's law given 

 in Verdet's Theorie mecanique de la Chaleur, in Miiller's Lehrbuch der 

 Physik, and even in Wiedemann's Galvanismus. Now the product of 

 force and velocity is not work, but work divided by time — that is to say, 

 rate-of- working, or "power." This may account for the widely-spread 

 fallacy. Jacobi makes another curious slip in the memoir above alluded 

 to (p. 463), by supposing that the strength of the current can only 

 become =0 when the motor runs at an infinite speed. We all know now 

 that the current will be reduced to zero when the counter electromotive 

 force of the motor equals that of the external supply; and if this is finite, 

 the velocity of the motor, if there is independent magnetism in its magnets, 

 need also only be finite. This error — also to be found in Verdet — seems 

 to have thrown the latter off the track of the true law of efficiency, and 

 to have made him fall back on Jacobi's law. 



