Subtracting this sum from the preceding, there remains the sum of the momenta acting on the zone of the half-wheel from the exterior to the interior = 2_b_² _a_ - 4_b_ _a_² + 8/3_a_³ - 2(_b_ - λ)² (_a_ - λ) + 4(_b_ - λ) (_a_ - λ)² - 8/3(_a_ - λ)³ - 2_b_² λ - 4_b_ _a_ λ + 4_a_² λ - 2_a_ λ² + 2/3λ³ = 2λ (_b_(_b_ - _a_) - _b_ _a_ + 2_a_² - _a_λ + 1/3λ²) = 2λ ((_b_ - _a_)(_b_ - _a_) + _a_² - _a_λ + 1/3λ²) Then dividing this sum of the momenta by the sum of the pressure there will be 2λ(((_b_ - _a_)(_b_ - _a_) + _a_² - _a_λ + 1/3λ²)/(2λ(_b_ - _a_))) = _b_ + _p_ (_a_(_a_² - _a_λ + 1/3λ²)/(_b_ - _a_)) the distance of the center of the pressure from the level of the fluid, that is, to the distance of the result of all the pressure from that level. From this it is evident that the center of pressure falls under the center of the wheel, C, to the distance (_a_² - _a_λ + 1/3λ²)/(_b_ - _a_) . Whence multiplying this distance by the result of the pressure, or by 2λ(_b_ - _a_), we obtain 2λ(_a_² - _a_λ + 1/3λ²) to express the momentum of the horizontal pressure of the water, directed to make the wheel turn from L to O. Now the momentum with which the vertical impulse of the fluid tends to make the semicircle F C O L turn from O to L (supposing the wheel not with a simple zone, but with a circular plane) is = 2/3_a_³. Likewise the momentum of the impulse of the fluid to cause the internal semicircle V C I G from O to L is - 2/3(_a_ - λ)³. Then taking this second momentum from the first, the momentum of the zone from the fluid V G I O L F to give the wheel an impulse from O to L will be = 2/3(_a_³ - (_a_ - λ)³) = 2λ(_a_² - _a_λ + 1/3λ²) which is precisely the momentum with which the horizontal pressure of the fluid to impress on the wheel an impulse in the opposite direction, that is to say from L to O. Consequently from the pressure of the fluid the wheel cannot have any motion around its center. The weight of the wheel itself, by which the half-zone immersed in the water tends to make the wheel turn from L to O, and the half which is out of the water, to make it turn in the reverse direction, such a weight, I say, cannot induce any motion of rotation, and both halves remain in equilibrium around the center C. Article by William Nicholson William Nicholson was born in London in 1753; died in 1815. He was a scientist of note, and a writer of scientific subjects. In 1797 he established in London and continued publishing until 1814, a periodical entitled "Journal of Natural Philosophy, Chemistry and the Arts," known, however, throughout the civilized world as "Nicholson's Journal."