520 Notices respecting New Books. 
multiform representation of the same grouping we could. have 
affirmed: @ priori the number of groupings to be 2 x 3 x 2° or 48, 
whereas the true number is only 40. I believe that the above 
is the first instance of the doctrine of types making its appear- 
arice explicitly, and illustrated by example in the theory of taetie. 
It were much to be desired that some one would endeavour to 
collect and collate the various solutions that have been given of the 
noted'15-school-girl problem by Messrs. Kirkman (in the Ladies’ 
Diary), Moses Ansted (in the Cambridge and Dubtin Mathematical 
Journal), by Messrs. Cayley and Spottiswoode (in the Philosophi- 
cal Magazine and elsewhere), and Professor Pierce, the latest and 
probably the best (in the American Astronomical Journal), besides 
various others originating and still floating about in the fashion- 
able world (one, if not two, of which I remember having been com- 
municated to me many years ago by Mr. Archibald Smith, F.R.S.), 
with a view to ascertain whether they belong to the same or to 
distinct types of aggregation. 
——— 
LXXVII. Notices respecting New Books. 
A History of the Progress of the Calculus of Variations during the 
Nineteenth Century. By 1. Topuunter, M.A., Fellow and Prin- 
cipal Mathematical Lecturer of St. John's College, Cambridge. 
Cambridge: Macmillan and Co. 1861. 
R. ‘TODHUNTER, whose name is already so familiar to the 
mathematical student, has at length produced a work of much 
greater originality and research than any of his former and more ele- 
mentary treatises. 
The ‘Calculus of Variations,” one of the most difficult branches 
of pure mathematics, has been the subject of the labours of several 
eminent mathematicians, Euler, Lagrange, Gauss, Poisson, &c., 
whose successive researches and improvements form an exceedingly 
interesting department of scientific history, which, however, has 
hitherto been specially treated by only one writer in our own lan- 
guage, viz. Woodhouse, whose ‘Treatise on Isoperimetrical Problems 
and the Calculus of Variations’ was published in 1810, and is now 
an extremely scarce book. 
Woodhouse’s work has always received very high praise by such 
competent judges as Messrs. Peacock, Herschel, and Babbage, in their 
‘Examples ;’ Professor De Morgan, in his ‘ Differential and Inte- 
gral Calculus ;’ and Professor Jellett, in his ‘Calculus of Variations.’ 
But since its publication the calculus has been greatly advanced and 
improved ; and it is to record this progress that Mr. ‘l’odhunter has 
written the volume before us, which commences where Woodhouse 
left off. It is evidently the work of one who thoroughly understands 
the science itself, and who has most conscientiously and labo- 
riously consulted and studied all the available materials and sources 
of information. He unites the qualifications of a sound mathe- 
