236 



cause is influential to the extent of about 4 per cent. Hence the bird will not 

 preserve a horizontal flight, if the velocity falls below 100 feet per second, 

 without increasing its angle of flight more than the assumed 7°. Another 

 slight error occurs in Captain Hutton's calculations which is probably an over- 



30 



sight. Using his equation HE = ^ ^ — 



° ° 4 tan A E H + sin C E H . cos C E H 



he makes H E = 115, when A E H = 0, and C E H = 15°. The true result 

 is 120. 



When proper data have been obtained, the solution of the problems con- 

 nected with this " sailing flight " should, I think, be approached in an entirely 

 different manner from that adopted by Captain Hutton. His deductions as to 

 the resistance of the air to a projectile of the form of the albatros are of no 

 value at all, and may, I think, be shown to be inconsistent with facts already 

 ascertained. The principal portion of the resistance is that which is resolved 

 into a sustaining, or upward bearing, force, and this is exerted against the 

 obliquely exposed under surfaces of the bird. The formula for the resistance 

 of a fluid to a plane, moving obliquely through it is — 



R = 1 Q v 2 sin 3 I. A 



where Q is the density of the fluid, v the velocity of the stream plus that of 

 the plane if it is moving against it, I the Z of inclination to the stream, and 

 A the area of the plane. The two latter coefficients will have to be determined 

 from observations, which in the case of I it will be very difficult to make. If 

 the part of R which is resolved into a force sustaining the bird against gravity 

 be known, let this = C, then the retarding force of atmospheric resistance 

 against the inclined surfaces of the body and wings, — 



= 1 Q v 2 sin 3 I. A-C 



We must deduct this quantity from the whole retardation observed to find 

 what would be the resistance to the front surfaces of the bird when both body 

 and wings were horizontal. It is only by this process that we can obtain a 

 quantity which is comparable with the atmospheric resistance to round shot. 



Art. LX. — On Sinking Funds. By Captain F. W. Hutton, ECS. 



[Bead be/ore the Auckland Institute, September 7, 1868.] 



The subject of Sinking Funds is one of much importance to this and other 

 countries, but I have not been able to find any book that treats of it, and I 

 therefore think that an investigation of its principles may prove both useful 

 and interesting. 



By " Sinking Fund " is meant a sum of money put away annually in 

 order to pay off a loan. There are two principal ways in which this money 

 is applied : either it may be invested year by year until, with the interest 

 accruing on it, it amounts to a sum sufficient to pay off the original loan ; or 

 else it may be used to take up yearly a portion of the loan until the whole has 

 vanished. I propose to investigate both these methods, and then compare 

 them together. 



The second case, where the fund is applied yearly to buy up the loan, is 

 very simple. 



Let a equal the amount of the loan. 

 ,, p the amount put by as Sinking Fund each year, and 

 ,, T equal the number of years it will take to pay off the loan. Then 



