15 





18 



M to which it will be found that * can be Uken so (mall that the 

 aorta shall be convergent. Now of two series of the form AA " + uA - * ' 

 + . . . . the value of that in which m U the greater will diminish with- 

 out limit a* compared with the other, when A diminishes without 

 limit. rone.|iiently, overj- curre y=+*, which has ^'a-^'a. will 

 approach, before the point of contact U attained, nearer to y=+x than 

 any other in which +'a it not =f'o. Again, when +'a--+'a, thoie 

 CUM of y +je in which if"a = f "a, will approach nearer to jr f than 

 any in which f'a not =+"' ; * n<l "" Hence, to make y = f* 

 hare the closest possible contact withy = ^r when .r-o; give such 

 values to the constant* in y=+x as will satisfy ai many as possible of 

 the equation* fa = T, *'a = +'o,f'o = *'a, 4c. consecutively from the 

 beginning. This is a brief aketch, which can be filled up from any 

 elementary work ; and the following are the principal results : 



1. When the string of equations is satisfied up to f<"o = +l"'a, the 

 contact is said to be of the nth <>rd i . 



J. In contact of the nth order, the deflection <p{a + )i) <Ha + A) 

 dirainUhes with ', and vanishes in a finite ratio to it 



;i. In contact of an even order, the curves intersect at the point of 

 contact ; in contact of an odd order, they do not intersect at that 

 point. 



4. When curves have a contact of the nth order, no curve, having 

 with either a contact of an order inferior to the nth at the same point, 

 can pass between the two. 



5. A straight line, generally speaking, can have only a contact of the 

 first order with a curve ; and the equation to the tangent straight line 

 of the curvey=$ir, when .r- a, is y fa = ^"a(x a). But if it should 

 happen that f " = 0, f"a= 0, &c., up to $<*'a = 0, then for that point 

 the tangent has a contact of the nth order. Thus, at a point of con- 

 trary flexure the tangent has a contact of the second order, at least, 

 with the curve. 



0. A circle, generally speaking, can be made to have a contact of the 

 second order with a curve, and the equation of the most tangent circle, 

 or circle of CURVATURE, to the curve y=<w, at the point a;=o,is 



This circle cute the curve, generally speaking : if not, as for example, 

 at the vertices of an ellipse, it is evidence that the circle has a contact 

 of some higher and odd order. The centre of the circle of curvature is 

 a ]H,iut on the normal, being that at which the normal touches the 

 (volute. [iNVoi.rtE AND EVOLCIE.] 



N . >t only is the term tangent mort generally applied to the closest 

 straight line only, but frequently only to that portion of the straight 

 line which falls between the point of contact and the axis of ?. Again, 

 the normal is a straight line perpendicular to the tangent, drawn 

 through the point of contact : but this term also is frequently applied 

 only to that portion which falls between the point of contact and the 

 axis of x. It is with reference to this limitation that the terms sub- 

 tangent and subnormal are to be understood : the first meaning the 

 distance from the foot of the tangent to the foot of the ordinate ; 

 the second that from the foot of the ordinate to that of the normal. 

 The formula for the subtangent is <jxj-=-4>' ; that for the subnormal 

 <f>a x <t>'a. The sign determining the mode of taking the line from the 

 foot of the ordinate. 



Let be the angle made by the tangent with the axis of r ; usually 

 the angle made by that part of the tangent which has positive ordinal.".-* 

 with the positive side of the axis of r. Then /3, at the point whose 

 abscissa is x, is determined by the equation 



dv <ls rfy 



tan fl=;^.; and subtangent = yjj-, subnormal = y ,' . 



If we take the more general mode of measurement proposed in SKIN, 

 this equation remains equally true. Now, keeping strictly to that 

 mode, let !- the angle made by the tangent with the axis of .<, the 

 angle made by the radius vector r with the axis of r, and fi that made 

 by the tangent with the radius vector. It will be found, then, that 

 in all oases 



dt 



Unless the mode of attributing signs be carefully attended to, these 

 last equations, though always considered as universally true, are not 

 so in reality. 



TANGl'iy. A poisonous crystalline bitter substance found in the 

 Tanghin of Madagascar (Cerbera Tcnghin). Its composition is not 

 known, but it appears to contain no nitrogen. [CEHIU.KA T.VN<;IIIN, in 

 NAT. HIST. Div.J 



TANK. The sen** usually attached to this word, in England, in that 

 of a rather large vessel, for holding water or oth<-r li.|iiid, either 

 placed above or below the ground ; in India, the word is applied to the 

 receptacles formed for the purpose of storing rain-water, some ol 

 which have an area of 364 (Dilwara) or 437 acres (Kalingtir), with 

 enclosure dykes of 38 feet in height. 



Tanks intended to hold rain-water collected from houses, or the 

 drainage water of farm yards in agricultural districts, are rarely made 



if a greater capacity than 50 or 00 'cubic yards ; and they are con- 

 structed of the iinMt impermeable hydraulic masonry that can be 

 obtained ; but precaution* mimt always be taken that the materials so 

 mpl .yod i-li.niM not be of a nature to affect the chemical properties 

 of the waters. Brickwork set in Roman cement is, for instance. pi.- 

 Ferable for such uses to masonry of limestone, set in ordinary lime 

 mortar ; and a backing of good clay puddle in i made 



of concrete. Provision must be made f.. r ti\inK tin- xuetion pipes of 

 the pumps, for connecting the inlet and overflow pi|Nw, and for means 

 of access for examination and repair ; ventilation pipes must also be 

 provided. As it almost always happens that dead leaves, or other 

 organic impurities, are carried into tanks of the description under con- 

 sideration, it is advisable, if the water be required for domestic use, 

 to provide some means of filtration for the water, and this object 

 is effected in many very ingenious manners, two of which may here 

 be mentioned. For instance, in the great chalk plateau of I 

 Normandy the rain water tanks are sunk into the boulder clay, or tin: 

 chalk, as the case may be, and they are made of sufficient dimensions 

 to allow of the formation of a subsidiary lateral chamber, shut off 

 from the body of the tank by a cress wall of a porous material, 

 through which the water from the larger chamber filters into the sub- 

 sidiary one ; the suction pipe being placed in the latter, and the inlet 

 pipes in the larger chamber. The other description of rain water 

 la-ik is used in Venice, and it is formed by firstly making a Urge water- 

 tight enclosure, usually circular in plan, and then an inner, con 

 wall is built, leaving a large annular space between them : this annular 

 space is filled in to a certain height with filtering materials of 

 gravel, &c., through which the water passes until it reaches a series 

 of holes made in the lower portion of the inner wall ; it then 

 accumulates in this species of well, from whence it is drawn by 

 means of buckets. It must, however, always be observed that rain 

 waters so stored are not of a proper quality for human consumption, 

 and that they should be exclusively applied to washing, or to analogous 

 household purposes. 



The large vessels used for storing oils, or for the various operations 

 of manufacturing chemistry, are made of iron, wood lined with lead, 

 of zinc, or of other materials according to the nature of the liquids to 

 be contained. The principles of their construction are derived from 

 the ordinary laws of HTDUOSTATICS. 



As to the Indian catch-water tanks it may be observed that they are 

 in fact KKSEIIVOIRK, and the remarks made under that head will apply 

 to them. Generally speaking the tanks in question are principally 

 used for irrigation, and their dams are formed of earth-work 

 towards the up stream with stone, either laid dry, or with mortar. 

 They are provided with overflow dams, escape weirs, sluices, and all 

 the ordinary appliances of reservoirs; and they would appeir to have 

 been constructed upon the same principles at the present century as 

 they were in the remotest antiquity. Much information on this sub- 

 ject is to be found in Genrr.il Uaird Smith's works on ' Irrigation,' .HI. I 

 in the ' Selections from the Records of the Bombay Governmct 

 in the 'Selections from Public Correspondence," published by 

 authority of the late East India Company. 



Agricultural Tanks are sometimes large open receptacles, or ponds, 

 formed by excavating the ground and disposing the removed earth in 

 the form of banks to retain the water ; but the tanks which will here 

 be especially treated of are the smaller covered reservoirs used to 

 collect and retain liquid manure. 



These tanks are usually constructed of an oblong shape, of brick well 

 cemented, with one or more divisions, and capable of c. 

 least ton tunes as many hogsheads as there are heads of cattle on tin 

 farm. They are vaulted over, having a small aperture, in which a 

 pump is placed, sufficient to allow a man occasionally to clear out the 

 sediment, when the liquid has been pumped up. The form pr<-, 

 is that of a cube, or rather that of several cubes in succession. A tank 

 for a farm of 200 acres of arable land should be 15 feet wide. 1ft deep, 

 and 45 long, giving 3 cubes of 15 feet, or a cavity capable of containing 

 upwards of 10,000 cubic feet of liquid. In this tank the urine in 

 diluted with water to prevent too rapid decomposition, and also to 

 retain the ammonia which is formed ; for which purpose gypsum and 

 sulphate of copper are sometimes put into the tanks. In very porous 

 soils the bottom and sides must be puddled, to keep in the liquid : ..nd 

 it may be advantageous to build the walla in cement altogether. The 

 liquid from the yards and stables is carried into the tank by a main 

 drain constructed of brick or stone, and which receives a number of 

 smaller drains from every part of the yards and cattle-sheds. Thus 

 the litter in the yard is always dry, and none of the richness of the 

 manure i lost, l.y evaporation. 



Sometimes the tank is vaulted like a cellar under the cow-house and 

 stables, which are washed out twice every day, and all the du: 

 water are swept into a cess-pool communicating with the tank. A 

 \.-i \ diluted, but rich liquid soon tills the first division of the tank : a 

 sluice is then shut, and the next washings run into a second division, 

 and when that is full, into a third. In the meantime the c<- 

 of the first tank have undergone a certain fermentation, by which 

 the ammonia first evolved has become mild and impregnates the 

 water. It is then in a fit state to be carried on the land in tubs or 

 water-carte. When properly diluted, it accelerates vegetation in a 

 surprising degree; but if put on fresh, it burns the grass or any 



