308 Notices respecting New Boohs. 



directly to such a function as e*, our author prefers to start with 

 the definition " that a point is said to have logarithmic motion on a 

 straight line when its distance from a fixed point on the line is 

 equally multiplied in equal times " (p. 78) ; and then, from the discus- 

 sion that follows, the function e t , as it were, emerges. It is needless 

 to add that it is most instructive to see how familiar matters are 

 regarded by a mind of great acuteness. Other instances might be 

 given, such as the articles headed " Exact Definition of Tangent," 

 which leads up to another, entitled " Exact Definition of Velocity." 

 However instructive many of these discussions may be, it is never- 

 theless somewhat difficult to account for the presence of many 

 things in the volume. For instance, it is hard to see why a trea- 

 tise on Kinematics should contain a demonstration of the fact that 

 the flux of a function of functions is given by the formula 



and the more as the demonstration is hardly perfect. What Pro- 

 fessor Clifford says is this:— it denoting /(#, y), where x and y 

 are functions of t, we find 



"and when we strike out common factors and omit suffixes in this 

 last expression, it becomes osb x f-\-yb y f, where / has been shortly 

 wTitten instead of f(oo, y). Or, substituting u for/, we have" the 

 above formula (p. 66). We will not urge that the reader might 

 easily misunderstand the direction to strike out common factors on 

 the right-hand side of the equation. But we would ask how, in the 

 case of any function not purely algebraical, does the student know 

 that there is a common factor to be struck out of the numerator 

 and denominator of any one of the four fractions concerned ? The 

 fact is, that in all but a few cases the calculation in question can 

 hardly be performed except by the use of limits or infinitesimals. 



The point which has given rise to these remarks by no means stands 

 alone. We find in different parts of the volume : — accounts of 

 the elementary properties of quadric curves (pp. 27-31, 38, 39, 91) 

 and surfaces (pp. 172-176, 177-181); the method of finding the 

 fluxion of t n (p. 55), of finding the area of a parabola (p. 73), of 



establishing the relation I t k dt=t k + 1 : Jc-\-l (p. 73), and of pro- 



rw= 



ving that " the central projection of a harmonic range is also a har- 

 monic range "(p. 42). These and other things like them fairly suggest 

 the question, Eor what class of readers is the book designed ? The 



* Looking at this well-known relation when thus written, we are 

 strongly tempted to suspect that in matters of notation there is a good 

 deal of mere fashion or arbitrary preference. Mr. Clifford's way of writing 

 the formula combines two modes of notation, both of which, a few years 

 ago, would have been considered to have had their day. There is, how- 

 ever, a reason why the fluxional notation should not be altogether lost 

 sight of. 



