INFLECTION. 



morning the cloth was found reduced 

 nearly to a cinder, and the wood of the 

 chest completely charred in the inside. 

 An experiment was immediately made to 

 ascertain the true cause : a piece of the 

 same cloth was dipped in the same sort 

 of oil, and shut up in a box, and in three 

 hours it was found scorching hot, and on 

 opening the box it burst into a flame. 

 Hence the spontaneous combustion of 

 wool, or woollen yarn, which has some- 

 times happened when large quantities 

 have be,en kept in heaps without the ac- 

 cess of fresh air. The oil with which it is 

 dressed seems to be the chief cause of 

 combustion. Wheaten flour and charcoal 

 reduced to powder, and heated in large 

 quantities, have been known to take fire 

 spontaneously. 



The cases of the spontaneous human 

 combustion have never been satisfactori- 

 ly accounted for ; the facts themselves 

 seem to be well authenticated ; two are 

 recorded in the Philosophical Transac- 

 tions, and referred to under COMBUSTION. 

 They ought, however, to hold out a lesson 

 of warning to those habitually given to 

 excess with regard to spirituous liquors ; 

 for, in every case, the subjects of this ter- 

 rible calamity were drunkards, whose 

 favourite liquor was alcohol, in the shape 

 of brandy, gin, &c. 



INFLECTION, or point of infection, in 

 the higher geometry, is the point where 

 a curve begins to bend a contrary way. 

 See FLEXURE. , 



There are various ways of finding the 

 point of inflection; but the following 

 seems to be the most simple. From the 

 nature of curvature it is evident, that 

 while a curve is concave towards an axis, 

 the fluxion of the ordinate decreases, or 

 is in a decreasing ratio, with regard to the 

 fluxion of the absciss ; but, on the contra- 

 ry, that the said fluxion increases, or is in 

 an increasing ratio to the fluxion of the 

 absciss, where the curve is convex to- 

 wards the axis ; and hence it follows that 

 those two fluxions are in a constant ratio 

 at the point of inflection, where the curve 

 is neither concave nor convex. That is, 

 if x = the absciss, and y = the ordinate 

 then x is to y in a constant ratio, or 



or^ is a constant quantity. But con- 

 y x 



stant quantities have no fluxion, or their 

 fluxion is equal to nothing; so that in 



this case the fluxion of or of -is equal 



y x H 



to nothing, And hence we have this 

 general rule : viz. put the given equa- 

 tion of the curve into fluxions ; from 



which equation of the fluxions find either 



or - ; then take the fluxion of this ratio 

 y x 



or fraction, and put it equal to or no- 

 thing ; and from this last equatian find 



also the value of the same or -. : then 



y x 



put this latter value equal to the former, 

 which will be an equation, from whence, 

 and the first given equation of the curve, 

 x and y will be determined, being the 

 absciss or ordinate answering to the point 

 of inflection in the curve. Or, putting the 



fluxion of r equal to 0, that is-. y "" a X y - 



= 0, or x y x y = 0, or x y = x y,' 

 or x : y : : x : y, that is, the second flux- 

 ions have the same ratio as the first flux- 

 ions, which is a constant ratio ; and there- 

 fore if a- be constant, or x = 0, then shall 

 y'be =0 also ; which gives another rule, 

 viz. take both the first and second flux- 

 ions of the given equation of the curve, in 

 which make both x and y = 0, and the re- 

 sulting equations will determine the va- 

 lues of x and y, or absciss and ordinate, 

 answering to the point of inflection. 



To determine the point of inflection in 

 curves, whose semi-ordinates C M, C m 

 (Plate Miscel. VII. fig. 13 and 14.) are 

 drawn from the fixed point C ; suppose 

 C M to be infinitely near C m, and make 

 m H = M m ; let T m touch the curve in 

 M. Now the angles C m T, C Mm, are 

 equal ; and so the angle C m H, while the 

 semi-ordinates increase, does decrease, if 

 the curve is concave towards the centre 

 C, and increases, if the convexity turns 

 towards it. Whence this angle, or, which 

 is the same, its measure, will be a mini- 

 mum or maximum, if the curve has a point 

 of inflection or retrogression ; and so 

 may be found, if the arch T H, or fluxion 

 of it, be made equal to 0, or infinity. 

 And in order to find the arch T H, draw 

 m L, so that the angle T m L be equal to 

 m C L ; then if C m = y y m r = x, mT 



= t, we shall have y : x \ : t : . Again, 



draw the arch H O to the radius C H ; 

 then the small right lines m r, O H, are 

 parallel ; and so the triangles O L H, m 

 L r, are similar; but because HI is also 

 perpendicular to m L, the triangles L H I. 

 m r, are also similar : whence i-.x-.-.y, 



xy 



; that is, the quantities m T, m L, are 



equal. But H L is the fluxion of H r, 

 which is the distance of C m = y ; and 



