a Truncated Triangular Pyramid. 175 



relation) in 0, let us understand that ABCO, BCDO, &c. form so 

 many harmonic systems of points ; B may be then called a har- 

 monic centre of AC, A and C opposites to B ; also we may call AB, 

 BC harmonic steps of the succession, so that by multiplying a 

 line AB n times, or making AX equal to n times AB, we are con- 

 structing the point X to which A will be transferred by n har- 

 monic steps, of which AB is the first ; and by ra-secting a line 

 AX, we mean finding a point B in it such that a succession of n 

 harmonic steps, commencing with AB, will carry A to X. 



In all this there is of course nothing new : those principles 

 are familiar to all geometers, and have received their fullest de- 

 velopment at the hands of Professor Cayley. We know a priori 

 that the descriptive properties included in the preceding (or 

 similar) constructions, such, e. g. } as that the six cross-triangles 

 of a frustum all meet in a point, will remain true when, adopting 

 a fixed plane of relation, we substitute harmonic centres in respect 

 to that plane in lieu of arithmetical centres*. Or, again, we may 

 affirm that the lines joining the harmonic centres of the opposite 

 edges of a tetrahedron will all intersect and harmonically bisect each 

 other, and so on. But what is further wanted, and what I will 

 proceed to supply, is a firm quantitative basis to this enlarged 

 theory, so formed as that we shall be able in the general case to 

 follow step by step the reasoning used in the common theory 

 where the plane of relation goes off to infinity, and to assign to 

 every point determined in the general constructions as distinctive 

 a character as it possesses in the special ones. This may be done 

 by the aid of very elementary considerations, which I proceed to 

 unfold, and which will be seen at once to bring the general or 

 perspective theory under .the dominion of the so-called integral 

 calculus or calculus of continuity. 



The arithmetical centre of two points A, B is the centre of 

 gravity of two equal atoms at A and B; let us then so assign 

 the weights of the atoms A, B in the general case as to make 

 their centre of gravity fall on the harmonic centre : this may 

 evidently be done by considering their weights as proportional 

 to their inverse distances from the plane of relation, and accord- 

 ingly we shall understand by the weight of an atom at any point 

 a quantity proportional to its inverse distance from the plane of 

 relation. But, moreover, the centre of gravity of the homoge- 

 neous line AB ought to fall at this same point, which we may if 

 we please consider as an inference at the limit from the same 



* Geometers have long been familiar with the idea of the pole or har- 

 monic centre of a triangle in respect to a line in its plane ; the principles 

 now about to be developed will enable us to attach a precise significa- 

 tion to the pole or harmonic centre of every geometrical figure of any form 

 whatever. 



