Notices respecting Neiv Hooks. ' 233 



aby giving different values to r different invariants can be obtained 

 e -or a given form; and then the questions arise, are they indepen- 

 T lent of each other ? if not, which are the fundamental invariants, 

 'and in what way are the others derived from them ? E. g. in the 

 case of the quintic form (n=5) there are four fundamental inva- 

 riants, from which the others can be deduced by a comparatively 

 simple process ; the four, however, are not independent of each 

 other, for it is found that the square of the invariant of the 18th 

 degree can be expressed in terms of those of the 4th, 8th, and 

 12th degrees. 



An enormous extension of the subject is due to the fact that 

 when a form (<p) is given, it is possible to assign a second form (\jj) 

 whose coefficients are connected with those of the given form in 

 such a manner that, when both undergo linear transformation, the 

 same relation exists between the coefficients of the transformed 

 functions as between the coefficients of the functions in their ori- 

 ginal shape ; <p and \j/ are called covariants. In the work before 

 us the subject of Covariants occupies more than twice the space de- 

 voted to Invariants, though the limitation to Binary forms is 

 strictly observed. 



It is this limitation which will render the work of great value to 

 students, as it enables the author to give a complete exposition of 

 the elements of the subject. To use his own words : — " There is 

 nothing better for the studious reader who wishes to make progress 

 than to have recourse to the original memoirs themselves ; and this 

 he will be able to do with safety and advantage when he has 

 thoroughly learned all that I have expounded in the present work " 

 (p. vi) . Within the assigned limits the subject is worked out with 

 ' great completeness : thus, not only are the properties of Symmetrical 

 functions of the Roots of Equations demonstrated and the means 

 by which their calculation can be expedited explained, but in an 

 Appendix the results of the actual calculation up to the eleventh 

 degree are given. In a second Appendix are given the invariants 

 of forms up to the fifth degree, expressed both as functions of the 

 coefficients and as functions of the roots, and in an abridged shape 

 those of forms of the sixth degree. In the same Appendix the 

 covariants are given of forms from the third up to the sixth degree. 

 It would seem that M. de Bruno has performed all the needful 

 calculations himself, though some at least of the results had been 

 published without his being aware of it at the time his calculations 

 were made. In short no pains have been spared to render the work 

 as complete as possible ; and, indeed, it appears from an incidental 

 remark (p. vi) that the author must have had the work on hand for 

 something like twenty years. 



The notation employed is of necessity complicated; and this, 

 while it makes misprints misleading to the reader, makes them also 

 hard to avoid. Though the book is printed in a good type and has 

 altogether a good appearance, we cannot help fearing that a minute 

 examination would yield a large crop of errata ; at all events we 

 'have come across several which are not noticed in the list at the 



