Apeil 16, 1897.] 



SCIENCE. 



603 



" There are things called Algebraical 

 Forms ; Professor Caj'ley calls them Quali- 

 ties. These are not, properly speaking, 

 Geometrical Forms, although capable, to 

 some extent, of being embodied in them, 

 but rather schemes of processes, or of opera- 

 tions for forming, for calling into existence, 

 as it were, algebraic quantities. 



" To every such Quantic is associated an 

 infinite variety of other forms that may be 

 regarded as engendered from and floating, 

 like an atmosphere, around it ; but infinite 

 in number as are these derived existences, 

 these emanations from the parent form, it is 

 found that they admit of being obtained by 

 composition, by mixture, so to say, of a cer- 

 tain limited number of fundamental forms, 

 standard rays, as they might be termed, in 

 the Algebraic Spectrum of the Quantic to 

 which they belong ; and, as it is a leading 

 pursuit of the physicists of the present day 

 to ascertain the fixed lines in the spectrum 

 of every chemical substance, so it is the aim 

 and object of a great school of mathemati- 

 cians to make out the fundamental derived 

 forms, the Covariants and Invariants, as 

 they are called, of these Quantics. 



" This is the kind of investigation in 

 which I have, for the last month or two, 

 been immersed, and which I entertain great 

 hopes of bringing to a successful issue. 



"Why do I mention it here ? It is to illus- 

 trate my opinion as to the invaluable aid 

 of teaching to the teacher, in throwing him 

 back upon his own thoughts and leading 

 him to evolve new results from ideas that 

 would have otherwise remained passive or 

 dormant in his mind. 



"But for the persistence of a student of this 

 university in urging upon me his desire to 

 study with me the modern algebra I should 

 never have been led into this investigation ; 

 and the new facts and principles which I 

 have discovered in regard to it (important 

 facts, I believe) would, so far as I am con- 

 cerned, have remained stiU hidden in the 



womb of time. In vain I represented to 

 this inquisitive student that he would do 

 better to take up some other subject lying 

 less off the beaten track of study, such as 

 the higher parts of the Calculus or Elliptic 

 Functions, or the theory of Substitutions, 

 or I wot not what besides. He stuck with 

 perfect respectfulness, but with invincible 

 pertinacity, to his point. He would have 

 the IvTew Algebra (Heaven knows where he 

 had heard about it, for it is almost unknown 

 on this continent), that or nothing. I was 

 obliged to yield, and what was the conse- 

 quence ? In trying to throw light upon an 

 obscure explanation in our text-book my 

 brain took fire; I plunged with requickened 

 zeal into a subject which I had for years 

 abandoned, and found food for thoughts 

 which have engaged my attention for a con- 

 siderable time past, and will probably oc- 

 cupy all my powers of contemplation ad- 

 vantageously for several months to come." 



Another specific instance of the same 

 thing he mentions in his paper, ' Proof of 

 the Hitherto Undemonstrated Fundamental 

 Theorem of Invariants,' dated November 

 13, 1877: 



" I am about to demonstrate a theorem 

 which has been waiting proof for the last 

 quarter of a century and upwards. It is 

 the more necessary that this should be 

 done, because the theorem has been sup- 

 posed to lead to false conclusions, and its 

 correctness has consequently been im- 

 • pugned. Thus in Professor Faa de Bruno's 

 valuable Theorie des formes binaires, Turin, 

 1876, at the foot of page 150 occurs the 

 following passage: " Cela suppose essen- 

 tiellement que les equations de condition 

 soient toutes independahtes entr^elles, ce qui n'est 

 pas toujours le cas, ainsi qu'il resulte des re- 

 cherches du Professor Gordan sur les nom- 

 bres des covariants des formes quintique 

 et sextique." 



The reader is cautioned against suppos- 

 ing that the consequence alleged above does 



