278 REMARKS ON SOME GENERAL PROPERTIES OF CURVES. 
REMARKS ON SOME GENERAL PROPERTIES OF CURVES. 
BY J. W. MARTIN, LL.D. 
THe geometric method of investigation, so highly esteemed by 
Newton and his followers, has experienced considerable vicissitude as 
regards the amount of attention bestowed upon it by mathematicians 
at different periods. Having for more than a century held undisputed 
sway in the universities of Great Britain, it was at length obliged to 
yield to those more powerful methods of investigation, which, prose- 
cuted with untiring zeal and ingenuity by men possessing unrivalled 
powers of analysis, had placed the continental mathematicians so far 
in advance of those in England. Though for a time decried as much 
as it was before injudiciously extolled, the geometric method has 
never been utterly neglected. It possesses merits of its own that 
must ever claim the attention of men of science. It affords solutions 
of many questions far more concise than can be furnished by the ana- 
lyst, and occasionally presents us with theorems which, as beautiful 
as unexpected, shew that its powers have not even yet been developed 
to the utmost. 
1. If two curves lie, the one inside the other, and a right line be 
drawn cutting the curves so that the sum of the areas of the segments 
cut off shall be constant, the envelop of the right line is the locus of 
the centre of gravity of the sum of the chords. 
2. Similarly, if the difference of areas is constant the envelop of 
line is locus of centre of gravity of difference of chords, that is of 
the portions of the right line enclosed between the two curves. 
These theorems have been slightly altered in form so as to exhibit 
more strongly an analogy to a theorem given by Professor Cherriman, 
in the Canadian Journal, February, 1863. 
3. The envelop of chords cutting a curve at equal angles is locus of 
a point dividing these chords, so that rectangle under segments is 
constant. 
4. The envelop of chords joining points of taction of parallel tan- 
gents is locus of a point dividing those chords in a given ratio. 
If the curve is a central conic the envelop is a point, the centre of 
coniessii 
5. If the curves S and S' are go related that tangent at any point 
