An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

Xvi SUMMARY.
ARTICLE PAGRK
341. To find the equation to the normal plane, or the plane
which is perpendicular to the direction of a curve of
double curvature at any point...................... 485
342. To find a surface of given species which shall osculate
with a curve of double curvature at any point......... 486
CHAP. XV.
Miscellaneous Theorems and Problems.
344. It is required to prove the following formula:
m m ((m - ) - 1)( _2) &c.
MM L(m - $+ - (m - &
continued to m terms = 1.2.3. &c m.................. 489
345. To investigate the law of the formation of the coefficients
of the terms of the Multinomial Theorem, by means of the
Differential Calculus.........4................... 490
346. It is required to decompose m --- into a series of fractions having simple or quadratic denominators......... 491
347. To find the sum of n terms of the series
12 sin + 32 sin 30 + sin 5 &............ 492
348. It is required to find the form of the functionf, so that
f (x + ) +f (x -- h) =f (x)f (h)........... 493
349. To find the particular value of the fraction
m am
(F x) n+ (F x)- +- &c.
(f x)n + ( f2X) + &c.
when such a value is assigned to x, as causes each of the
terms to vanish...494
351. To find the shortest straight line that can be drawn
through a given point, between two other straight lines
at right angles to each other.......................... 495
352. Of all triangles having a given base and vertical angle,
to find that whose area is the greatest...............496
353. To find the radius of a circle such that the segment of it,
formed by an arc of given length, may be the greatest
possible....................................... 497