327 
of Edinburgh, Session 1879-80. 
The theory is applied to the logarithmic and circular functions, 
and at the end of Part I. are given some very ingenious applications 
to expansions in fractional powers of x. 
In Part II. is given the following extremely elegant formula — 
J d6(J>(6 + a)(z- 0) p = (- l) p+1 !p+ 1 cos(^+ 0( 2 . + a )» 
which is applied to the solution of a variety of problems. 
The whole of the mathematical work in this memoir is of great 
simplicity and elegance, and for that reason alone it is well worth 
the attention of students of the higher mathematics. It has, more- 
over, intrinsic value as an important contribution to the elucidation 
of a difficult branch of analysis. How great that importance may 
be it is impossible to estimate until the future of the method is more 
certain than it can at present be said to be ; but, in any case, the 
work will remain a lasting monument to the skill and ingenuity of 
its author. 
Closely connected with the paper just mentioned is another on a 
process in the differential calculus and its application to the solution 
of differential equations. Nothing farther need be said regarding 
it except that it is characterized by the same elegance and simplicity 
that mark the memoir on general differentiation. 
Perhaps the most important of all Professor Kelland’s scientific 
papers is his Memoir on the limits of our knowledge respecting the 
Theory of Parallels. He there deals with the subject now better 
known as absolute or non-Euclidean geometry. It would scarcely be 
possible to convey to those who have not busied themselves with pan- 
geometry (or the geometry of pure reason as one might venture to 
call it, as opposed to the geometry of experience which is Euclid’s) a 
full idea of the importance of this work of Kelland’s, and of the 
evidence that it affords of his grasp of purely mathematical specula- 
tion. Suffice it to say that he reasons out correctly, and perhaps even 
more elegantly than is done in one of the last works on the subject,* 
the consequences of denying Euclid’s “parallel axiom,” or what is its 
equivalent, viz., the proposition that the sum of the three angles of 
any triangle is two right angles. It can be shewn by means of the 
properties of congruent figures, which, with all the consequences as 
* Frischauf, “ Elemente der Absolute!! Geometrie.” 
