to the simmaiion of certain extensive classes of Series. SS 
and making all the necessary substitutions and reductions we 
have, 
where, being even, y’(^) must be an even function, and the 
last term of (p (d) will be (2). ; but if n be odd then f (^) 
must be an odd function, and ” (S). will be the last 
term. This is in fact precisely the theorem demonstrated in 
Phil. Trans. 1814. 
Although when the series (p ^ is susceptible of such a sum 
as that under contemplation, the above investigation will inva- 
riably give it ; yet the results it leads to (though no apparent 
error can be detected in the analysis,) are in some cases at va- 
riance with those afforded by other methods, and this indicates 
the necessity of some criterion to distinguish a priori whether 
any contradiction is involved in supposing it so susceptible. It 
is easy to assign forms to f(J), (such for instance as 
which shall be alike susceptible of developement in positive and 
negative powers of and it is requisite in many cases to take 
into consideration the negative developement in order to get the 
true sum. But to enter into any discussion on this very interest- 
ing part of the subject, would far exceed the proper limits of 
the present essay. 
January % 1819. 
Art. VII. — On the Application of Oars to Steam-Boats. By 
Mr Richard Whytock, Edinburgh. 
It is impossible to observe the motion and the progress of the 
latest constructed of our Steam Boats, without being convinced, 
that the art of navigation by means of such vessels is yet in its 
infancy. The disadvantages attending the present mode of ap- 
plying the power to the water, for the purpose of propelling the 
vessel, seem to be very generally acknowledged ; but no more 
effectual method has yet been put in practice. 
VOL. II. NO. 3. JANUARY 1820. C 
