Conditions Imposed by a Set of Order Numbers 145 



number ri less y.^^ ^ , the order number of the factor ((2, «))^ with a branch 

 of the second cycle. We thus impose on the function ((s,?^)) further 

 conditions (r2— /z^, 2)^2 — 2 W — l + l/''2) "2 in number because of the 

 alternative arbitrary character of the factor ((2, «))„_;, _i . 

 We now have 



21. ((z,Z^)) = ((2,w)),^((z,iO).2((2>w))„_,^_,^_l . 



Here we may assume ((2, m))^ to coincide with 72(2, u) out to a point such 

 that its orders of coincidence with the branches of the cycles other than 

 the second are the same as the orders of coincidence of /2(3, u) with these 

 branches. Also the above factorization does not impose on the function 

 ((z, «)) an order number for the 1st cycle which is greater than n, 

 because of the alternative possibility referred to above in which the 

 factor {{z,uy)y does not contribute the total order number ri. 



Continuing the process of factorization by which we arrived at (21) 

 the splitting of a factor ((z,w));,^ from ((2,w))„_^^_^2_i imposes on the 

 coefficients of the function ((2,w)) additional conditions (^3 — ^1,3— y-i,^v-i — 

 I (^13'— l + l/j'3)f3 in number. So proceeding we ultimately arrive 

 at a factorization of the type 



((2, m)) = ((s, m)),^((z, w)).2 • • • ((z, w)),^_j ((2, U)),^- 1 



and determine thereby for the number of the conditions imposed on the 

 function ((2, «)) by the order numbers ri, . . . . , r^ the expression 



r r t — \ r 



2 TtVi- S 2 y.s,iVt — \ 2 {ili-\-\-\lv^Vi 

 which is evidently equal to 



r r 



2 Ttvt—h 2 {ni—l-\-l/vt)vt. 



In what precedes we have not gone into detail as much as we might 

 have done. We have not followed all the leads which presented them- 

 selves. We have by no means exhausted all the lines of proof which 

 were suggested by the context. What we have done is to illustrate in 

 a humble way what the writer recently heard called the method of 

 "nosing along" by one who in reading a mathematical book had noted 

 therein the employment of this method and had accordingly decided 

 against its scientific merits because he had failed to grasp the fact that 

 this is the one and only method for the systematic uncovering of truth 

 vouchsafed to man and compatible with his psychology. 



