July 22, 1909] 



NA TURE 



where there is much traffic requires re-covering about 

 once in four years. Instances are given from the 

 author's own experience where roads having a fair 

 amount of traffic " have been as good at the end of 

 six years as when first covered, and so far as can be 

 seen will need very little for another six years." 



This book ought to be carefully studied by all 

 surve)'ors having charge of ro.ids subject to motor 

 traffic. 



VECTORIAL GRAPHICS. 

 Vectors and Vector Diagrams applied to the Alter- 

 nating Current Circuit. By W. Cramp and C. F. 

 Smith. Pp. xvi + 252. (London : Longmans, Green 

 and Co., 1909.) Price 7s. 6d. net. 



NOT many years ago a certain type of journalist 

 used to compare and contrast the theorist and 

 the practical man, to the demolition of the former 

 and the apotheosis of the latter. Fortunately, such 

 an attitude of mind is no longer possible. The merely 

 practical man could never have constructed the Forth 

 Bridge, nor launched the Mauritania on her record- 

 making career. Innumerable examples might be 

 given of the necessity of true theory in the economical 

 designing of all kinds of machinery ; but probably 

 there is nothing that better proves how much mathe- 

 matical science lies at the foundation of modern 

 methods than electrical applications, especially those 

 that have to do with the alternating current. The 

 whole history of the development of the transformer 

 and the alternating-current motor is simply the 

 realisation of the solution of a differential equation 

 given long ago by Maxwell. In this realisation the 

 first great steps were taken by Heaviside, who in- 

 troduced the terms impedance, admittance, reluctance, 

 &c., giving a new precision to the ideas involved. By 

 a mathematical extension of meaning the symbols 

 which entered into the electrical equations of steady 

 currents became applicable to the corresponding cases 

 of sinusoidal currents. Stated in purely mathematical 

 language, this transition depended on the properties 

 of the complex variable. 



Thus, to take the simplest case, Ohm's law RC = E 

 for steady electromotive force becomes Maxwell's 

 expression (R + Lii/dt)C = E when E is variable. 

 Representing a sinusoidal electromotive by the ex- 

 ponential of the imaginary ipt, we get the solution in 

 the form (R + iL/))C = E, where C and E now stand 

 for the amplitudes of the varying quantities. This 

 complex quantity which operates on C may be treated 

 analytically like the real quantity R in Ohm's law. 

 Multiplication by the conjugate gives 



( R- + L'7t-)C = ( R - /L/> )E. 



In the end, after all analytical transformations have 

 been effected, the real part of the expression must be 

 picked out. A little experience will make the average 

 student quite efficient in this kind of algebra, 

 especially if it is combined with numerical and 

 practical work. 



But the value of the method does not stop here. 

 Following familiar paths, we may give a geometrical 

 form to the expressions, and obtain graphical repre- 

 NO. 2073, VOL. 81] 



sentations of important relations. Thus the complex 

 quantity RC + iLpC may be laid down as a vector in 

 a plane, RC being the component along a chosen 

 direction and hpC the component at right angles to 

 this direction, while the ratio L/)/R measures the 

 tangent of the angle between the vector and the 

 chosen direction of reference. Again, if we regard C 

 as a complete vector, the complex operator may be 

 considered to be a versor rotating C through the angle 

 just named. Can we utilise these fundamental 

 vectorial and versorial conceptions to construct a 

 graphical representation of real value to the electrical 

 engineer? The answer has been given in the affirma- 

 tive ; and among those who have worked up the 

 method along these lines, no one holds a higher place 

 than C. P. Steinmetz. The method has been pre- 

 sented in more or less detail in most of the recent 

 books on the alternating current, and now we have 

 an extremely valuable addition to the literature of the 

 subject in " Vectors and Vector Diagrams applied to 

 the Alternating Current Circuit," the joint work of 

 William Cramp and Charles F. Smith, both lecturers 

 in the electrical engineering school of Manchester 

 University. The authors, for reasons given, depart 

 somewhat from Steinmetz in their development of the 

 method, but the foundation is essentially the same. 

 Once the fundamental propositions are admitted and 

 grasped, the whole treatment is a model of lucidity 

 and self-consistency. One unusual feature of the 

 book is that it assumes a certain fairly advanced 

 knowledge at the start. This is a good feature, which 

 might well characterise more of our text-books. The 

 authors are careful at the same time to indicate 

 exactly what knowledge the student must possess 

 before he is in a position to make effective use of their 

 methods — he must know the fundamental laws of the 

 alternatmg-current circuit very thoroughly. Neverthe- 

 less, it would have been of advantage to have indi- 

 cated in a few preliminary sections the manner in 

 which the method originally took shape as a synthesis 

 of the symbolic solutions of Maxwell's differential 

 equations. There also seems to be a certain looseness 

 of argument in the way in which the properties of 

 vectors are stated. For example, having defined in 

 the usual geometrical way the meaning of the " vector 

 product " of two vectors, and having so named it, 

 they remark, " This product must itself be a vector 

 product, since it has already been shown to possess a 

 definite sense." This is no proof, but mere statement. 

 The defined product must be shown to obey the vector 

 law of addition before it can be called a vector product. 

 These imperfections do not, however, affect the 

 purpose of the authors, who are to be congratulated 

 on having enriched our technical literature with a 

 clear and systematic exposition of the vectorial 

 graphics of alternating-current phenomena. After a 

 discussion of the more purely geometrical character of 

 the method, illustrated throughout by reference to 

 familiar electrical phenomena, a succession of chapters 

 follows on self and mutual induction, the transformer, 

 motors of the induction type, and alternating-current 

 commutator motors. A chapter is then thrown in on 

 the product of two vectors, and the two concluding 

 and longest chapters deal respectively with locus 



