294 SCIENCE PROGRESS 



What is the relation between 6'3 and the incomplete result ot 3*2 ? On 

 effecting the iteration indicated in 6 - 2, we have by 4*1 



<£« = m + {<t>'mY . (O - m) + ^.T^S ' — ^ m > ' x • (O - w) 2 + etc. 



(d>'m) n — i)<b"m 

 = m- (4>'m) n m + K -^~ '- ±-— (<h'm) n - 1 m 2 + etc. + terms in O. 6' 4 



Now in 3'2 we saw that <£ 3 , as we attempted to evolve it by substitutions, gives a 

 cumbersome series consisting only of the constants a, 6, c, . . . followed by un- 

 ascertained terms in ascending powers of O. This cumbersome series, then, must 

 be the same as the terms in m indicated above. To show their identity we must 

 therefore find an algebraic expression for m from the equation (f>m = m. We 

 proceed as follows : 



a + im + cm*+ . . .=m, 



[« + (*- 1)0 + ^0* + . . .>» = o, 

 m = [a + (6-i)0 + cO* + . . .]-\o) = [[a + 0][(6 T i)0 + cO i + .. .]]" 1 (o) 

 = [(£-i)0 + <:0 J +. . .y^O-aJo) (since [Vxl'^X" 1 ^" 1 ) 

 = -(b-i)-ia-(6-i)- i ca i -{(6-i)- 1 d-2(d-i)-*c i }(6-i)- s a*-etc, 6'$ 



by putting n— — I in 4/1, and remembering that [O - a](o) = -a. The same value 

 of m can be obtained by " operative division " (see my papers 1 and 4). By 

 substituting it in 6*3 we repeat the series commenced in 3*2. 



7*0. A word now on the geometrical interpretation of the above. As suggested 

 in my paper (4), the Cartesian expression of curves may be abbreviated by denoting 

 them by operations alone, apart from the subjects— a thing which the present 

 notation allows us to do* Thus we express the curve y — <px by 4> alone ; a + do is 

 the straight line ; and O is the midaxis, usually written y—x. Obviously m denotes 

 an intersection of <j> an d O, a point of which the co-ordinates are y = m, x = m, 

 where m is a real root of (f>m-m-o, that is, m = [<f> -0] _1 (o). I call m a mid- 

 axial root of <j>, as distinct from its axial roots. 



The curve [fi + 0]tj or^ + | is a curve similar to £ but translated upward to a 

 distance p along the 000, or y, axis; and the curve |[0-?], or $(0-g), is one 

 similar to $ but translated rightward to a distance q along the oO, or x, axis ; or, 

 what is the same thing, these expressions may denote the transference of the 

 origin by the same distances reversed. Hence [w + 0]£[0-w] is the curve $ 

 shifted to a distance m both upward and rightward, simultaneously, without rotation. 



The curve bO + cO* + . . . passes through the origin, where it has the tangential 

 slope b ; and it possesses a midaxial root y — o, x = o, and can be iterated by 4"i. 

 The curve <f>~a + l>0 + cO*+ ■ ■ . does not pass through the origin ; and in order 

 to iterate it by the method of 6'2, we find a midaxial root of it (if it possesses one), 

 namely y =*n, x=m, and then we find a curve $ which is the same as <£ but trans- 

 lated so as to pass through the origin and be iterable by 4*1 . To do this, we have 

 by 6'2, £ = [w-r-0] _1 <£[>» + 0]. Then after iterating | we pass back to <p n by the 

 retransformation indicated in 6'2, that is, 



<£• = [ m + o]£"|> + O] " \ 

 Of course there are many curves {e.g. e') which have no real midaxial roots at all. 

 Others have more than one real midaxial root. 



If we draw the graph of <f> and ^, it is easy to draw graphs of 4>yjr, and 

 therefore of <£*, <£ 3 , . . . and of <£~\ ^ _1 , etc. It is also easy to make graphs of the 

 iterations denoted by <px, <ftx, . . . <f> n x, and to show how these iterations may 

 under certain conditions approximate to alternate midaxial roots of <f> denoted by 



* Necessarily an inadequate description. 



